From d695eaacccedfc979f3d5624b0f57320defff474 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 17:49:48 +0400 Subject: [PATCH 01/39] chore: add refs.bib with Duchi et al. (2008) entry Co-Authored-By: Claude Sonnet 4.6 --- paper/bib/refs.bib | 200 ++++++++++++++++++++++++++ src/fast_minimum_variance/proximal.py | 0 2 files changed, 200 insertions(+) create mode 100644 paper/bib/refs.bib create mode 100644 src/fast_minimum_variance/proximal.py diff --git a/paper/bib/refs.bib b/paper/bib/refs.bib new file mode 100644 index 0000000..4681f47 --- /dev/null +++ b/paper/bib/refs.bib @@ -0,0 +1,200 @@ +@incollection{simoncini2013, + author = {Simoncini, Valeria and Szyld, Daniel B.}, + title = {On the Superlinear Convergence of {MINRES}}, + booktitle = {Numerical Mathematics and Advanced Applications 2011}, + publisher = {Springer}, + year = {2013}, + pages = { }, + note = {Proceedings of {ENUMATH} 2011}, +} + +@book{fischer1996, + author = {Fischer, Bernd}, + title = {Polynomial Based Iteration Methods for Symmetric Linear Systems}, + publisher = {Wiley-Teubner}, + year = {1996}, +} + +@article{bunch1977, + author = {Bunch, James R. and Kaufman, Linda}, + title = {Some stable methods for calculating inertia and solving symmetric linear systems}, + journal = {Mathematics of Computation}, + year = {1977}, + volume = {31}, + number = {137}, + pages = {163--179}, +} + +@book{boyd2004, + author = {Boyd, Stephen and Vandenberghe, Lieven}, + title = {Convex Optimization}, + publisher = {Cambridge University Press}, + year = {2004}, +} + +@article{benzi2005, + author = {Benzi, Michele and Golub, Gene H. and Liesen, Jörg}, + title = {Numerical solution of saddle point problems}, + journal = {Acta Numerica}, + year = {2005}, + volume = {14}, + pages = {1--137}, +} + +@article{bajeux2012, + author = {Bajeux-Besnainou, Isabelle and Bandara, Wimal and Bura, Efstathia}, + title = {A {K}rylov subspace approach to large portfolio optimization}, + journal = {Journal of Economic Dynamics and Control}, + year = {2012}, + volume = {36}, + number = {11}, + pages = {1688--1699}, +} + +@article{clarabel2024, + author = {Goulart, Paul J. and Chen, Yuwen}, + title = {Clarabel: An interior-point solver for conic optimisation problems with + quadratic objectives}, + journal = {INFORMS Journal on Computing}, + year = {2024}, + volume = {36}, + number = {2}, + pages = {481--499}, +} + +@article{diamond2016, + author = {Diamond, S. and Boyd, S.}, + title = {{CVXPY}: A {Python}-embedded modeling language for convex optimization}, + journal = {Journal of Machine Learning Research}, + year = {2016}, + volume = {17}, + number = {83}, + pages = {1--5}, +} + +@misc{fast_minimum_variance, + author = {Schmelzer, T.}, + title = {fast-minimum-variance: Solving Minimum Variance Portfolios Fast}, + howpublished = {\url{https://github.com/Jebel-Quant/fast_minimum_variance}}, + year = {2026}, + note = {{MIT} License}, +} + +@book{golub2013, + author = {Golub, Gene H. and Van Loan, Charles F.}, + title = {Matrix Computations}, + edition = {4th}, + publisher = {Johns Hopkins University Press}, + year = {2013}, +} + +@article{ledoit2004, + author = {Ledoit, O. and Wolf, M.}, + title = {A well-conditioned estimator for large-dimensional covariance matrices}, + journal = {Journal of Multivariate Analysis}, + year = {2004}, + volume = {88}, + number = {2}, + pages = {365--411}, +} + +@article{ledoit2020, + author = {Ledoit, O. and Wolf, M.}, + title = {Analytical nonlinear shrinkage of large-dimensional covariance matrices}, + journal = {Annals of Statistics}, + year = {2020}, + volume = {48}, + number = {5}, + pages = {3043--3065}, +} + +@article{merton1972, + author = {Merton, Robert C.}, + title = {An Analytic Derivation of the Efficient Portfolio Frontier}, + journal = {Journal of Financial and Quantitative Analysis}, + year = {1972}, + volume = {7}, + number = {4}, + pages = {1851--1872}, +} + +@article{markowitz1952, + author = {Markowitz, H.}, + title = {Portfolio selection}, + journal = {Journal of Finance}, + year = {1952}, + volume = {7}, + number = {1}, + pages = {77--91}, +} + +@article{hestenes1952, + author = {Hestenes, Magnus R. and Stiefel, Eduard}, + title = {Methods of conjugate gradients for solving linear systems}, + journal = {Journal of Research of the National Bureau of Standards}, + year = {1952}, + volume = {49}, + number = {6}, + pages = {409--436}, +} + +@article{paige1975, + author = {Paige, Christopher C. and Saunders, Michael A.}, + title = {Solution of sparse indefinite systems of linear equations}, + journal = {SIAM Journal on Numerical Analysis}, + year = {1975}, + volume = {12}, + number = {4}, + pages = {617--629}, +} + +@book{greenbaum1997, + author = {Greenbaum, Anne}, + title = {Iterative Methods for Solving Linear Systems}, + publisher = {SIAM}, + address = {Philadelphia}, + year = {1997}, + series = {Frontiers in Applied Mathematics}, + volume = {17}, +} + +@phdthesis{choi2006, + author = {Choi, Sou-Cheng T.}, + title = {Iterative Methods for Singular Linear Equations and Least-Squares Problems}, + school = {Stanford University}, + year = {2006}, + type = {PhD thesis}, +} + +@unpublished{minvar_paper, + author = {Schmelzer, Thomas and Wolf, Michael and Stoll, Martin}, + title = {Krylov Methods for the Minimum Variance Portfolio}, + note = {Working paper, Jebel Quant}, + year = {2026}, +} + +@phdthesis{schmelzer2004, + author = {Schmelzer, Thomas}, + title = {Block {K}rylov methods for {H}ermitian linear systems}, + school = {Department of Mathematics, University of Kaiserslautern}, + year = {2004}, + type = {Diploma thesis}, +} + +@inproceedings{duchi2008, + author = {Duchi, John and Shalev-Shwartz, Shai and Singer, Yoram and Chandra, Tushar}, + title = {Efficient Projections onto the $\ell_1$-Ball for Learning in High Dimensions}, + booktitle = {Proceedings of the 25th International Conference on Machine Learning ({ICML})}, + year = {2008}, + pages = {272--279}, + publisher = {ACM}, + address = {New York, NY}, +} + +@book{lawson1974, + author = {Lawson, Charles L. and Hanson, Richard J.}, + title = {Solving Least Squares Problems}, + publisher = {Prentice-Hall}, + year = {1974}, + address = {Englewood Cliffs, NJ}, +} diff --git a/src/fast_minimum_variance/proximal.py b/src/fast_minimum_variance/proximal.py new file mode 100644 index 0000000..e69de29 From 0387d6923c14a8daa82ac1c0ada5923c4f2e06d3 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 17:53:56 +0400 Subject: [PATCH 02/39] test: add test suite for proximal module Co-Authored-By: Claude Sonnet 4.6 --- tests/conftest.py | 12 ++ tests/test_proximal.py | 270 +++++++++++++++++++++++++++++++++++++++++ 2 files changed, 282 insertions(+) create mode 100644 tests/conftest.py create mode 100644 tests/test_proximal.py diff --git a/tests/conftest.py b/tests/conftest.py new file mode 100644 index 0000000..aea27ef --- /dev/null +++ b/tests/conftest.py @@ -0,0 +1,12 @@ +"""Shared pytest fixtures.""" + +from __future__ import annotations + +from pathlib import Path + +import pytest + + +@pytest.fixture(scope="session") +def resource_dir() -> Path: + return Path(__file__).parent / "resources" diff --git a/tests/test_proximal.py b/tests/test_proximal.py new file mode 100644 index 0000000..1e49fb0 --- /dev/null +++ b/tests/test_proximal.py @@ -0,0 +1,270 @@ +"""Tests for the proximal module.""" + +from __future__ import annotations + +import numpy as np +import pytest + +from fast_minimum_variance.proximal import proj_simplex, prox_gradient + + +class TestProjSimplex: + """Test suite for proj_simplex function.""" + + @pytest.mark.parametrize("n", [2, 5, 10, 100, 1000]) + def test_output_sums_to_one(self, n: int) -> None: + """Verify projected vector sums to 1 (default radius).""" + rng = np.random.default_rng(42) + vec = rng.standard_normal(n) + result = proj_simplex(vec) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-10) + + @pytest.mark.parametrize("rad", [0.5, 1.0, 2.0, 10.0]) + def test_output_sums_to_radius(self, rad: float) -> None: + """Verify projected vector sums to specified radius.""" + rng = np.random.default_rng(42) + vec = rng.standard_normal(10) + result = proj_simplex(vec, rad=rad) + np.testing.assert_allclose(result.sum(), rad, rtol=1e-10) + + def test_output_non_negative(self) -> None: + """Verify all elements of projected vector are non-negative.""" + rng = np.random.default_rng(42) + vec = rng.standard_normal(100) + result = proj_simplex(vec) + assert np.all(result >= 0) + + def test_idempotent(self) -> None: + """Projecting a valid simplex point should return the same point.""" + vec = np.array([0.2, 0.3, 0.5]) + result = proj_simplex(vec) + np.testing.assert_allclose(result, vec, rtol=1e-10) + + def test_already_on_simplex(self) -> None: + """Vector already on simplex should be unchanged.""" + vec = np.array([0.25, 0.25, 0.25, 0.25]) + result = proj_simplex(vec) + np.testing.assert_allclose(result, vec, rtol=1e-10) + + def test_single_element(self) -> None: + """Single element vector projects to [rad].""" + result = proj_simplex(np.array([5.0])) + np.testing.assert_allclose(result, np.array([1.0]), rtol=1e-10) + + def test_all_negative_input(self) -> None: + """All negative inputs should project to corner of simplex.""" + vec = np.array([-1.0, -2.0, -3.0, -4.0, -5.0]) + result = proj_simplex(vec) + # Should project to [1, 0, 0, 0, 0] since -1 is largest + assert result[0] == pytest.approx(1.0, rel=1e-10) + assert np.sum(result[1:]) == pytest.approx(0.0, abs=1e-10) + + def test_uniform_input(self) -> None: + """Uniform input should project to uniform simplex point.""" + vec = np.array([3.0, 3.0, 3.0, 3.0]) + result = proj_simplex(vec) + expected = np.array([0.25, 0.25, 0.25, 0.25]) + np.testing.assert_allclose(result, expected, rtol=1e-10) + + @pytest.mark.parametrize("n", [10, 100, 1000]) + def test_projection_is_closest_point(self, n: int) -> None: + """Verify projection is the closest point on simplex.""" + rng = np.random.default_rng(42) + vec = rng.standard_normal(n) + proj = proj_simplex(vec) + + # Any point on simplex should be farther from vec than proj + # Test with some random simplex points + for _ in range(10): + random_simplex = rng.dirichlet(np.ones(n)) + dist_to_proj = np.linalg.norm(vec - proj) + dist_to_random = np.linalg.norm(vec - random_simplex) + assert dist_to_proj <= dist_to_random + 1e-10 + + def test_two_elements(self) -> None: + """Test projection with only 2 elements.""" + vec = np.array([2.0, -1.0]) + result = proj_simplex(vec) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-10) + assert np.all(result >= 0) + + def test_large_values(self) -> None: + """Test projection with very large values.""" + vec = np.array([1e10, 1e10, 1e10]) + result = proj_simplex(vec) + # Numerical precision degrades with very large values + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) + np.testing.assert_allclose(result, np.array([1 / 3, 1 / 3, 1 / 3]), rtol=1e-5) + + def test_small_values(self) -> None: + """Test projection with very small values.""" + vec = np.array([1e-10, 2e-10, 3e-10]) + result = proj_simplex(vec) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-10) + + def test_mixed_scales(self) -> None: + """Test projection with mixed scale values.""" + vec = np.array([1e8, 1e-8, 1e4, 1e-4]) + result = proj_simplex(vec) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-10) + assert np.all(result >= 0) + + +class TestProxGradient: + """Test suite for prox_gradient function.""" + + def test_cla(self, resource_dir) -> None: + """Test against CLA expected values.""" + covar = np.genfromtxt(resource_dir / "CLA_Data.csv", delimiter=",", skip_header=1)[3:] + result = prox_gradient(covar, np.ones(10)) + expected = np.array([0, 0.41200, 0, 0, 0, 0, 0.27612, 0.31188, 0, 0]) + assert np.linalg.norm(result - expected, 1) < 1e-5 + + def test_output_on_simplex(self) -> None: + """Verify output satisfies simplex constraints.""" + rng = np.random.default_rng(42) + mat = rng.standard_normal((5, 3)) + vec = rng.standard_normal(5) + result = prox_gradient(mat, vec) + + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-6) + assert np.all(result >= -1e-10) + + @pytest.mark.parametrize("shape", [(3, 2), (5, 5), (10, 3), (20, 10)]) + def test_various_shapes(self, shape: tuple[int, int]) -> None: + """Test with various matrix shapes.""" + rng = np.random.default_rng(42) + m, n = shape + mat = rng.standard_normal((m, n)) + vec = rng.standard_normal(m) + result = prox_gradient(mat, vec) + + assert result.shape == (n,) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) + assert np.all(result >= -1e-10) + + def test_identity_matrix(self) -> None: + """Test with identity matrix - solution should be projection of vec.""" + n = 5 + mat = np.eye(n) + vec = np.array([0.1, 0.2, 0.3, 0.2, 0.2]) # Already on simplex + result = prox_gradient(mat, vec) + np.testing.assert_allclose(result, vec, rtol=1e-5) + + def test_convergence_tolerance(self) -> None: + """Test that tighter tolerance gives valid results.""" + rng = np.random.default_rng(42) + mat = rng.standard_normal((10, 5)) + vec = rng.standard_normal(10) + + result_loose = prox_gradient(mat, vec, eps_rel=1e-3) + result_tight = prox_gradient(mat, vec, eps_rel=1e-10) + + # Both should satisfy constraints + np.testing.assert_allclose(result_loose.sum(), 1.0, rtol=1e-3) + np.testing.assert_allclose(result_tight.sum(), 1.0, rtol=1e-10) + + def test_max_iter_respected(self) -> None: + """Test that max_iter parameter is respected.""" + rng = np.random.default_rng(42) + mat = rng.standard_normal((100, 50)) + vec = rng.standard_normal(100) + + # Should still produce valid output even with few iterations + result = prox_gradient(mat, vec, max_iter=10) + assert result.shape == (50,) + + def test_objective_convergence(self) -> None: + """Test that algorithm converges to similar objective values.""" + rng = np.random.default_rng(42) + mat = rng.standard_normal((5, 3)) + vec = rng.standard_normal(5) + + # Run multiple times - results may vary due to random init + # but objective values should be similar + objectives = [] + for _ in range(5): + result = prox_gradient(mat, vec, eps_rel=1e-10) + obj = 0.5 * np.linalg.norm(mat @ result - vec) ** 2 + objectives.append(obj) + + # All objectives should be close to the minimum + np.testing.assert_allclose(objectives, objectives[0], rtol=1e-3) + + def test_near_zero_matrix(self) -> None: + """Test with near-zero matrix (low Lipschitz constant).""" + mat = np.array([[1e-10, 0], [0, 1e-10]]) + vec = np.array([1.0, 1.0]) + result = prox_gradient(mat, vec) + assert result.shape == (2,) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) + + def test_rank_deficient(self) -> None: + """Test with rank-deficient matrix.""" + mat = np.array([[1, 2, 3], [2, 4, 6]]) # Rank 1 + vec = np.array([1.0, 2.0]) + result = prox_gradient(mat, vec) + assert result.shape == (3,) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) + + +class TestNumericalStability: + """Test numerical stability with challenging inputs.""" + + def test_ill_conditioned(self) -> None: + """Test with ill-conditioned matrix.""" + # Create an ill-conditioned matrix + u = np.array([[1, 0], [0, 1], [0, 0]]) + s = np.array([1e6, 1e-6]) + vt = np.array([[1, 0], [0, 1]]) + mat = u @ np.diag(s) @ vt + vec = np.array([1.0, 1.0, 1.0]) + + result = prox_gradient(mat, vec, eps_rel=1e-6, max_iter=5000) + assert result.shape == (2,) + # May not converge perfectly but should satisfy constraints approximately + assert abs(result.sum() - 1.0) < 0.01 + + @pytest.mark.parametrize("seed", range(5)) + def test_random_problems(self, seed: int) -> None: + """Test with various random problem instances.""" + rng = np.random.default_rng(seed) + m, n = rng.integers(5, 20), rng.integers(2, 10) + mat = rng.standard_normal((m, n)) + vec = rng.standard_normal(m) + + result = prox_gradient(mat, vec) + np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) + assert np.all(result >= -1e-10) + + +class TestKKTConditions: + """Test mathematical properties via KKT conditions.""" + + @pytest.mark.parametrize("seed", range(5)) + def test_proj_simplex_kkt(self, seed: int) -> None: + """Verify KKT optimality conditions for simplex projection. + + For the projection problem: + min 0.5 ||x - v||^2 + s.t. sum(x) = 1, x >= 0 + + KKT conditions require: + - Primal feasibility: sum(x) = 1, x >= 0 + - For active constraints (x_i > 0): x_i = v_i - lambda + """ + rng = np.random.default_rng(seed) + v = rng.standard_normal(10) + x = proj_simplex(v) + + # Primal feasibility + np.testing.assert_allclose(x.sum(), 1.0, rtol=1e-10) + assert np.all(x >= -1e-12) + + # For x_i > 0, we have x_i = v_i - lambda + # So lambda should be the same for all active components + active_mask = x > 1e-10 + if np.any(active_mask): + lambdas = v[active_mask] - x[active_mask] + # All should be approximately equal + np.testing.assert_allclose(lambdas, lambdas[0], rtol=1e-8) From fa82490e82ca02beb76da8b0d890b30287d39ae6 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 17:58:12 +0400 Subject: [PATCH 03/39] feat: add solve_proximal to _BaseProblem Co-Authored-By: Claude Sonnet 4.6 --- src/fast_minimum_variance/_base.py | 43 ++++++++++++++++++++++++++++++ 1 file changed, 43 insertions(+) diff --git a/src/fast_minimum_variance/_base.py b/src/fast_minimum_variance/_base.py index 5ca6fb2..4d77883 100644 --- a/src/fast_minimum_variance/_base.py +++ b/src/fast_minimum_variance/_base.py @@ -284,6 +284,49 @@ def solve_clarabel(self, *, project: bool = True): w = self._clip_and_renormalize(w) return w, sol.iterations + def solve_proximal(self, *, project: bool = True): + """Solve via proximal gradient descent projected onto the probability simplex. + + Minimises ``0.5 * w^T Σ_LW w`` subject to ``w >= 0, sum(w) = 1`` by + casting the objective as a least-squares problem and applying the + Duchi et al. (2008) simplex-projection step. Return tilt (``rho != 0``) + is not supported. + + Args: + project: Clip and renormalize after solving (see ``solve_kkt``). + + Returns: + ``(w, 1)`` — weight vector of shape ``(N,)`` and iteration count + (always 1; the inner iteration count is not exposed). + + Examples: + >>> import numpy as np + >>> from fast_minimum_variance import Problem + >>> X = np.random.default_rng(0).standard_normal((100, 5)) + >>> w, iters = Problem(X).solve_proximal() + >>> float(round(w.sum(), 10)) + 1.0 + >>> bool((w >= 0).all()) + True + """ + from .proximal import prox_gradient + + if self.target is None: + mat = self.X / np.sqrt(self.t) + vec = np.zeros(self.t) + else: + chol_t = np.linalg.cholesky(self.target) + mat = np.vstack([ + np.sqrt(1.0 - self.alpha) / np.sqrt(self.t) * self.X, + np.sqrt(self.alpha) * chol_t.T, + ]) + vec = np.zeros(mat.shape[0]) + + w = prox_gradient(mat, vec) + if project: + w = self._clip_and_renormalize(w) + return w, 1 + def solve_osqp(self, *, project: bool = True): """Solve via OSQP (operator-splitting QP solver, direct API, no CVXPY overhead). From da448ba6b13584121c37434eb4467b5cc5c54a98 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:01:09 +0400 Subject: [PATCH 04/39] test: add TestSolveProximal to test_proximal and test_base Co-Authored-By: Claude Sonnet 4.6 --- tests/test_base.py | 46 ++++++++++++++++++++++++++++ tests/test_proximal.py | 69 ++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 115 insertions(+) diff --git a/tests/test_base.py b/tests/test_base.py index 46acbf4..16dec17 100644 --- a/tests/test_base.py +++ b/tests/test_base.py @@ -220,3 +220,49 @@ def _cvxpy_constraints(self, w, cp_module): _SpyStub(_X3).solve_cvxpy() assert received["w_type"] == "Variable" assert received["cp"] is cp + + +# --------------------------------------------------------------------------- +# solve_proximal +# --------------------------------------------------------------------------- + + +class TestSolveProximal: + """Tests for _BaseProblem.solve_proximal template.""" + + def test_returns_tuple(self): + """solve_proximal returns a (w, iters) tuple.""" + result = _Stub(_X3).solve_proximal() + assert isinstance(result, tuple) and len(result) == 2 + + def test_iters_is_one(self): + """Iteration count is always 1.""" + _, iters = _Stub(_X3).solve_proximal() + assert iters == 1 + + def test_weight_shape(self): + """Returned weight vector has shape (N,).""" + w, _ = _Stub(_X3).solve_proximal() + assert w.shape == (_X3.shape[1],) + + def test_project_true_sums_to_one(self): + """project=True ensures weights sum to 1.""" + w, _ = _Stub(_X3).solve_proximal(project=True) + assert w.sum() == pytest.approx(1.0) + + def test_project_true_non_negative(self): + """project=True ensures all weights are non-negative.""" + w, _ = _Stub(_X3).solve_proximal(project=True) + assert np.all(w >= 0) + + def test_project_false_still_on_simplex(self): + """project=False skips clip-and-renormalize; prox_gradient already enforces simplex.""" + w, _ = _Stub(_X3).solve_proximal(project=False) + np.testing.assert_allclose(w.sum(), 1.0, rtol=1e-6) + assert np.all(w >= -1e-10) + + def test_project_default_clips_and_renormalizes(self): + """Default (project=True) clips and renormalizes like other template solvers.""" + w, _ = _Stub(_X3).solve_proximal() + assert w.sum() == pytest.approx(1.0) + assert np.all(w >= 0) diff --git a/tests/test_proximal.py b/tests/test_proximal.py index 1e49fb0..7413ad5 100644 --- a/tests/test_proximal.py +++ b/tests/test_proximal.py @@ -5,9 +5,14 @@ import numpy as np import pytest +from fast_minimum_variance import Problem from fast_minimum_variance.proximal import proj_simplex, prox_gradient +def make_returns(T, N, seed=0): # noqa: N803 + return np.random.default_rng(seed).standard_normal((T, N)) + + class TestProjSimplex: """Test suite for proj_simplex function.""" @@ -268,3 +273,67 @@ def test_proj_simplex_kkt(self, seed: int) -> None: lambdas = v[active_mask] - x[active_mask] # All should be approximately equal np.testing.assert_allclose(lambdas, lambdas[0], rtol=1e-8) + + +class TestSolveProximal: + """Cross-validation: solve_proximal vs CVXPY reference.""" + + @pytest.fixture(scope="class") + def X(self): # noqa: N802 + return make_returns(T=200, N=10, seed=42) + + @pytest.fixture(scope="class") + def X_small(self): # noqa: N802 + return make_returns(T=100, N=5, seed=7) + + def test_return_shape(self, X) -> None: # noqa: N803 + """solve_proximal returns (w, iters) with w of shape (N,).""" + w, iters = Problem(X).solve_proximal() + assert w.shape == (X.shape[1],) + assert iters == 1 + + def test_weights_sum_to_one(self, X) -> None: # noqa: N803 + """Weights must sum to 1.""" + w, _ = Problem(X).solve_proximal() + np.testing.assert_allclose(w.sum(), 1.0, rtol=1e-6) + + def test_weights_non_negative(self, X) -> None: # noqa: N803 + """Weights must be non-negative after projection.""" + w, _ = Problem(X).solve_proximal() + assert np.all(w >= 0) + + def test_plain_minvar_vs_cvxpy(self, X) -> None: # noqa: N803 + """Plain minimum variance agrees with CVXPY reference.""" + w_prox, _ = Problem(X).solve_proximal() + w_cvx, _ = Problem(X).solve_cvxpy() + np.testing.assert_allclose(w_prox, w_cvx, atol=1e-3) + + def test_with_shrinkage_vs_cvxpy(self, X) -> None: # noqa: N803 + """Shrinkage (alpha > 0, target = I) agrees with CVXPY reference.""" + T, N = X.shape # noqa: N806 + alpha = N / (N + T) + w_prox, _ = Problem(X, alpha=alpha, target=np.eye(N)).solve_proximal() + w_cvx, _ = Problem(X, alpha=alpha, target=np.eye(N)).solve_cvxpy() + np.testing.assert_allclose(w_prox, w_cvx, atol=1e-3) + + def test_small_problem_vs_cvxpy(self, X_small) -> None: # noqa: N803 + """Small problem (T=100, N=5) agrees with CVXPY reference.""" + w_prox, _ = Problem(X_small).solve_proximal() + w_cvx, _ = Problem(X_small).solve_cvxpy() + np.testing.assert_allclose(w_prox, w_cvx, atol=1e-3) + + @pytest.mark.parametrize("N", [2, 5, 20]) + def test_various_sizes(self, N) -> None: # noqa: N803 + """Agreement with CVXPY holds for several problem sizes.""" + X = make_returns(T=5 * N, N=N, seed=N) # noqa: N806 + w_prox, _ = Problem(X).solve_proximal() + w_cvx, _ = Problem(X).solve_cvxpy() + np.testing.assert_allclose(w_prox, w_cvx, atol=1e-3) + + def test_project_false(self, X) -> None: # noqa: N803 + """project=False skips clip-and-renormalize; result still on simplex from prox_gradient.""" + w_proj, _ = Problem(X).solve_proximal(project=True) + w_raw, _ = Problem(X).solve_proximal(project=False) + # Both should satisfy constraints (prox_gradient enforces simplex internally) + np.testing.assert_allclose(w_raw.sum(), 1.0, rtol=1e-6) + assert np.all(w_raw >= -1e-10) From 21ced499e3649af7703b61a18e77d307ecafa0c1 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:03:28 +0400 Subject: [PATCH 05/39] feat: add proximal solver to make_minvar_figures benchmarks and plots Co-Authored-By: Claude Sonnet 4.6 --- book/marimo/notebooks/make_minvar_figures.py | 20 ++++++++++++-------- 1 file changed, 12 insertions(+), 8 deletions(-) diff --git a/book/marimo/notebooks/make_minvar_figures.py b/book/marimo/notebooks/make_minvar_figures.py index d35418f..39953dd 100644 --- a/book/marimo/notebooks/make_minvar_figures.py +++ b/book/marimo/notebooks/make_minvar_figures.py @@ -57,6 +57,7 @@ def _(): ("kkt", lambda: MinVarProblem(R_sp).solve_kkt()), ("cg", lambda: MinVarProblem(R_sp).solve_cg()), ("nnls", lambda: MinVarProblem(R_sp).solve_nnls()), + ("proximal", lambda: MinVarProblem(R_sp).solve_proximal()), ] configs_sp_lw = [ ("cvxpy", lambda: MinVarProblem(R_sp, alpha=alpha_sp, target=target_sp).solve_cvxpy()), @@ -65,6 +66,7 @@ def _(): ("kkt", lambda: MinVarProblem(R_sp, alpha=alpha_sp, target=target_sp).solve_kkt()), ("cg", lambda: MinVarProblem(R_sp, alpha=alpha_sp, target=target_sp).solve_cg()), ("nnls", lambda: MinVarProblem(R_sp, alpha=alpha_sp, target=target_sp).solve_nnls()), + ("proximal", lambda: MinVarProblem(R_sp, alpha=alpha_sp, target=target_sp).solve_proximal()), ] sp_no_lw, sp_lw = {}, {} @@ -86,10 +88,10 @@ def _(): print("=" * 70) ns = [50, 100, 200, 300, 500, 750, 1000, 1500, 2000] - times = {k: [] for k in ("kkt", "cg", "nnls")} + times = {k: [] for k in ("kkt", "cg", "nnls", "proximal")} rng2 = np.random.default_rng(0) - print(f"{'n':>6} {'kkt':>10} {'cg':>10} {'nnls':>10}") - print("-" * 44) + print(f"{'n':>6} {'kkt':>10} {'cg':>10} {'nnls':>10} {'proximal':>10}") + print("-" * 57) for n in ns: T = 2 * n # noqa: N806 R = rng2.standard_normal((T, n)) # noqa: N806 @@ -99,10 +101,12 @@ def _(): _, t_kkt = run_timed(lambda r=R, a=alpha, t=tgt: MinVarProblem(r, alpha=a, target=t).solve_kkt()) _, t_cg = run_timed(lambda r=R, a=alpha, t=tgt: MinVarProblem(r, alpha=a, target=t).solve_cg()) _, t_nnls = run_timed(lambda r=R, a=alpha, t=tgt: MinVarProblem(r, alpha=a, target=t).solve_nnls()) + _, t_prox = run_timed(lambda r=R, a=alpha, t=tgt: MinVarProblem(r, alpha=a, target=t).solve_proximal()) times["kkt"].append(t_kkt) times["cg"].append(t_cg) times["nnls"].append(t_nnls) - print(f"{n:>6} {t_kkt:>10.4f} {t_cg:>10.4f} {t_nnls:>10.4f}") + times["proximal"].append(t_prox) + print(f"{n:>6} {t_kkt:>10.4f} {t_cg:>10.4f} {t_nnls:>10.4f} {t_prox:>10.4f}") # ── Panel B: iterations vs shrinkage intensity alpha ────────────────────────── @@ -125,9 +129,9 @@ def _(): fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(6.5, 2.8)) - colors = {"kkt": "#1f77b4", "cg": "#ff7f0e", "nnls": "#2ca02c"} - labels = {"kkt": "KKT direct", "cg": "CG (matrix-free)", "nnls": "NNLS"} - for key in ("kkt", "cg", "nnls"): + colors = {"kkt": "#1f77b4", "cg": "#ff7f0e", "nnls": "#2ca02c", "proximal": "#9467bd"} + labels = {"kkt": "KKT direct", "cg": "CG (matrix-free)", "nnls": "NNLS", "proximal": "Proximal gradient"} + for key in ("kkt", "cg", "nnls", "proximal"): ax1.plot(ns, times[key], marker="o", markersize=3, label=labels[key], color=colors[key]) ax1.set_xscale("log") ax1.set_yscale("log") @@ -156,7 +160,7 @@ def _(): fig2, ax = plt.subplots(figsize=(4.5, 3.2)) - for key in ("kkt", "cg", "nnls"): + for key in ("kkt", "cg", "nnls", "proximal"): ax.plot(ns, times[key], marker="o", markersize=4, label=labels[key], color=colors[key]) # Reference slope lines anchored to the KKT curve at n=500 From 3842052d0c7dece0e0414bc329a9fd4fba346cb9 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:07:35 +0400 Subject: [PATCH 06/39] chore: add proximal.py implementation and regenerated scaling figures Co-Authored-By: Claude Sonnet 4.6 --- .../marimo/notebooks/graphs/minvar_loglog.pdf | Bin 26241 -> 27170 bytes .../marimo/notebooks/graphs/minvar_loglog.png | Bin 84085 -> 90731 bytes .../notebooks/graphs/minvar_scaling.pdf | Bin 30705 -> 31686 bytes .../notebooks/graphs/minvar_scaling.png | Bin 94133 -> 103332 bytes src/fast_minimum_variance/proximal.py | 133 ++++++++++++++++++ 5 files changed, 133 insertions(+) diff --git 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z)7W#wZ@yiut@`OUYv3v8!OxHN%g@gQc4avJ9VnMR>uI~S^5Bf4f;ER7qxcTJ`YW<; z`i}2;OE%T~EJ|&Cev->upy!aC-tVLHwy|-GNdJ%6-UM7!0vgin2z{{u*cYvmFwGMA z_2uPa8Ow{S4d-%Qs=N#wGPx4g>XZSx)4oW=^?2y~+HW&&@izLcP?B6WdEWGwQzHca zH1_Ryu_qfi>zAb+q79$5HelH+#xHH=^Y-nz+}ItgvbM^XqYb_#&-e!%+tub3+>bbh)Rt3_2DH6ctFKbP62kS8`^~X`+qswBS3j3^P6= 0, sum(x) = 1}. + +This module implements proximal gradient descent for constrained linear least squares +optimization on the probability simplex. The algorithm is based on iterative projection +using the efficient simplex projection from Duchi et al. (2008). + +References: +---------- + +Duchi, J., Shalev-Shwartz, S., Singer, Y., & Chandra, T. (2008). +"Efficient Projections onto the l1-Ball for Learning in High Dimensions." +Proceedings of the 25th International Conference on Machine Learning (ICML). +""" + +from __future__ import annotations + +from typing import TYPE_CHECKING + +import numpy as np + +if TYPE_CHECKING: + from numpy.typing import NDArray + + +def proj_simplex( + vec: NDArray[np.floating], + rad: float = 1.0, +) -> NDArray[np.floating]: + """Project a vector onto the probability simplex. + + This function computes the Euclidean projection of a given vector onto the probability + simplex. The simplex is defined as the set of non-negative vectors that sum to a + given radius, typically 1. The projection ensures that the resulting vector satisfies + these constraints. + + The algorithm is based on Duchi et al. (2008) "Efficient Projections onto the + l1-Ball for Learning in High Dimensions". + + Parameters + ---------- + vec : NDArray[np.floating] + Input vector that is to be projected onto the simplex. + rad : float, optional + Radius of the simplex. The projected vector will have components summing + to this value. Default is 1.0. + + Returns: + ------- + NDArray[np.floating] + The projected vector that lies on the probability simplex. + + Raises: + ------ + ValueError + If the input vector is empty. + + Examples: + -------- + >>> import numpy as np + >>> vec = np.array([1.0, 2.0, 3.0]) + >>> result = proj_simplex(vec) + >>> bool(np.isclose(result.sum(), 1.0)) + True + >>> bool(np.all(result >= 0)) + True + + """ + muu = np.sort(vec)[::-1] + cummeans = 1 / np.arange(1, len(vec) + 1) * (np.cumsum(muu) - rad) + rho = max(np.where(muu > cummeans)[0]) + result: NDArray[np.floating] = np.maximum(vec - cummeans[rho], 0) + return result + + +def prox_gradient( + mat: NDArray[np.floating], + vec: NDArray[np.floating], + eps_rel: float = 1e-6, + max_iter: int = 1000, +) -> NDArray[np.floating]: + """Perform proximal gradient descent to solve a constrained optimization problem. + + Solves the optimization problem: + minimize 0.5 ||mat @ x - vec||^2 + subject to x >= 0, sum(x) = 1 + + The function uses proximal gradient descent with simplex projection to find + the solution. The step size is determined by the Lipschitz constant of the + gradient. + + Parameters + ---------- + mat : NDArray[np.floating] + A matrix of shape (n_samples, n_features) used in the optimization + problem. + vec : NDArray[np.floating] + A vector of shape (n_samples,) used in the optimization problem. + eps_rel : float, optional + The relative error threshold for stopping criteria. Default is 1e-6. + max_iter : int, optional + The maximum number of iterations for the algorithm. Default is 1000. + + Returns: + ------- + NDArray[np.floating] + The solution vector of shape (n_features,) obtained after the + optimization process. + + Examples: + -------- + >>> import numpy as np + >>> mat = np.array([[1.0, 0.5], [0.5, 1.0]]) + >>> vec = np.ones(2) + >>> result = prox_gradient(mat, vec) + >>> bool(np.isclose(result.sum(), 1.0)) + True + + """ + rng = np.random.default_rng() + prim_var: NDArray[np.floating] = np.asarray(rng.standard_normal(size=mat.shape[1])) + sym_mat = mat.T @ mat + lip = np.linalg.norm(sym_mat, 2) # Lipschitz constant of the gradient + step = 0.5 / lip if abs(lip) > 1e-15 else 1.0 + + out_prod = mat.T @ vec + ite = 0 + err_rel = eps_rel + 1 + while err_rel > eps_rel and ite < max_iter: + prim_var_new = proj_simplex(prim_var - step * (sym_mat @ prim_var - out_prod)) + err_rel = float(np.linalg.norm(prim_var - prim_var_new, 2)) + prim_var = prim_var_new.copy() + ite += 1 + return prim_var From d4eb3eddce2089f9e3afaeeda675c7111fb2c11f Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:12:20 +0400 Subject: [PATCH 07/39] docs: add Section 4.5 on proximal gradient solver to minvar_paper Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 774 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 774 insertions(+) create mode 100644 paper/minvar_paper.tex diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex new file mode 100644 index 0000000..a10769c --- /dev/null +++ b/paper/minvar_paper.tex @@ -0,0 +1,774 @@ +\documentclass{siam/siamart} + +\usepackage{amsmath, amssymb} +\usepackage{booktabs} +\usepackage{microtype} +\usepackage{xcolor} +\usepackage{graphicx} +\usepackage{algpseudocode} + +\newsiamremark{remark}{Remark} +\crefname{remark}{Remark}{Remarks} +% Fix siamart.cls's refstepcounter override: it strips \@currentcounter, +% which TeX Live 2026's kernel and cleveref's First Aid patch both require. +\makeatletter +\def\cref@old@refstepcounter#1{% + \stepcounter{#1}% + \edef\reserved@a{#1}% + \ifx\reserved@a\ltx@star@counter\else + \let\@currentcounter\reserved@a + \fi + \protected@edef\@currentlabel{% + \csname p@#1\expandafter\endcsname\csname the#1\endcsname}% +}% +\makeatother + + +\title{Krylov Methods for the Minimum Variance Portfolio} + +\author{Thomas Schmelzer% + \thanks{Jebel Quant Research (\email{thomas@jqr.ae})}% + \and + Michael Wolf% + \thanks{Department of Economics, University of Zurich.}% + \and + Martin Stoll% + \thanks{Faculty of Mathematics, TU~Chemnitz.}% +} + +\headers{Krylov Methods for the Minimum Variance Portfolio}{T.~Schmelzer, M.~Wolf, and M.~Stoll} + +\begin{document} + +\maketitle + +\begin{abstract} +The long-only minimum variance portfolio is routinely solved by assembling a sample covariance +matrix and dispatching it to a general-purpose quadratic programme. Both steps are unnecessary +and expensive. + +Conjugate gradients applied matrix-free to the returns matrix $X$ solve the problem in $O(nT)$ +work per iteration without ever forming the $n\times n$ covariance; a primal-dual active-set +loop enforces non-negativity with negligible overhead. The covariance matrix is not a primitive +of the problem --- it is a computational detour. + +Ledoit-Wolf shrinkage, already standard practice for statistical reasons, turns out to be +identical to spectral preconditioning of the linear system: the same ridge term that shrinks +sample eigenvalues toward their mean also compresses the condition number and cuts CG iteration +counts dramatically. Statistical regularisation and numerical preconditioning are the same +operation applied to the same matrix --- a connection that does not appear to have been made +explicit in the existing literature. + +Numerical experiments on S\&P~500 equity data and synthetic problems of growing size show +order-of-magnitude speedups over interior-point solvers. For standard portfolio sizes a direct +KKT solve already captures most of the gain; the matrix-free advantage of CG becomes decisive +at larger $n$, where avoiding explicit covariance assembly reduces per-step cost from $O(n^2)$ +to $O(Tn)$. +\end{abstract} + +\section{Introduction} + +The minimum variance portfolio allocates capital across $n$ assets so as to minimise return variance subject only to +full investment and non-negativity. Since \cite{markowitz1952} it has been standard practice to estimate a sample +covariance matrix $\hat\Sigma = X^\top X / T$ from a $T \times n$ return matrix $X$ and pass it to a general-purpose +quadratic programming solver. This paper argues that both steps are unnecessary and expensive. + +That the minimum-variance solution satisfies $w \propto \Sigma^{-1}\mathbf{1}$ is a classical observation: it follows +directly from the KKT stationarity condition $2\Sigma w = \lambda\mathbf{1}$ and was made explicit by +\cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the single rescaling +$w = v/(\mathbf{1}^\top v)$ where $v = \Sigma^{-1}\mathbf{1}$. What has \emph{not} been done is to exploit this +structure computationally: the standard pipeline still forms $\Sigma = X^\top X/T$ explicitly before solving. + +This paper makes two contributions. First, we apply conjugate gradients (CG) to $\Sigma v = \mathbf{1}$ +\emph{matrix-free}: $\Sigma = X^\top X$ is never assembled. Each CG iteration requires only two matrix-vector products +with $X$, reducing per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. Long-only weights are +enforced by a primal-dual active-set loop that calls the CG solver on successively smaller subproblems. + +Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004} plays a dual role in this framework. Statistically, it +reduces estimation error by shrinking sample eigenvalues toward their mean. Numerically, the same modification +compresses the eigenvalue spread of the system matrix and directly accelerates CG convergence: on S\&P~500 equity data, +shrinkage cuts CG iterations from 744 to 82. Statistical regularisation and spectral preconditioning are, in this +problem, the same operation applied to the same matrix---a connection that does not appear to have been made explicit in +the existing literature. + +CG~\cite{hestenes1952} is a cornerstone of large-scale scientific computing~\cite{golub2013} but has not been applied +to portfolio optimisation in the matrix-free form described here. The closest related work applies GMRES to the minimum +variance problem when $\Sigma$ is singular~\cite{bajeux2012}; that formulation still assembles $X^\top X$ and does not +exploit the SPD structure or consider preconditioning. General-purpose solvers such as +Clarabel~\cite{clarabel2024} handle the long-only constraint natively but require an explicit covariance matrix. +ywa +Sections~\ref{sec:problem}--\ref{sec:precond} develop the formulation and solvers; Section~\ref{sec:results} reports +empirical results on both synthetic and S\&P~500 data. + +\section{Problem Formulation}\label{sec:problem} + +Let $X \in \mathbb{R}^{T \times n}$ be the demeaned return matrix for $n$ assets over $T$ trading days. The long-only +minimum variance problem is: +\[ +\min_{w} \; w^\top \hat\Sigma\, w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, \quad w \geq 0 +\] +where $\hat\Sigma = \frac{1}{T} X^\top X$ is the sample covariance matrix. This is equivalently stated as: +\[ +\min_{w} \; \frac{\|Xw\|^2}{T} = \min_{w}\; w^\top \hat\Sigma\, w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, \quad w \geq 0 +\] +This reframing is key: the objective is now a normalised squared norm of $Xw$, and we can evaluate +$w \mapsto X^\top(Xw)$ using two matrix-vector products, without ever forming $X^\top X$. + +\begin{table}[ht] +\centering +\begin{tabular}{lll} +\toprule +Parameter & Type & Description \\ +\midrule +$X$ & $\mathbb{R}^{T \times n}$ & Demeaned return matrix \\ +$\texttt{target}$ & $\mathbb{R}^{n \times n}$ & Shrinkage target matrix (optional; default \texttt{None}) \\ +$\alpha$ & $[0,1]$ & Shrinkage intensity; active only when \texttt{target} is provided (default $0$) \\ +$\rho$ & $\mathbb{R}_{\geq 0}$ & Return-tilt weight (default $0$) \\ +$\mu$ & $\mathbb{R}^n$ & Expected returns (default $\mathbf{0}$) \\ +\bottomrule +\end{tabular} +\caption{Parameters of the \texttt{Problem} specification.} +\label{tab:params} +\end{table} + +\paragraph{Dual problem.} +Including multipliers for both constraints, the full Lagrangian is +\[ + \mathcal{L}(w, \lambda, \nu) + = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1) - \nu^\top w, +\] +where $\lambda\in\mathbb{R}$ is the budget multiplier and $\nu\geq\mathbf{0}$ is the vector of non-negativity +multipliers. Minimising over $w$ via stationarity $\frac{2}{T} X^\top X w = \lambda\mathbf{1}+\nu$ gives the dual function +\[ + g(\lambda,\nu) + = \inf_w \mathcal{L}(w,\lambda,\nu) + = \lambda - \tfrac{T}{4}(\lambda\mathbf{1}+\nu)^\top(X^\top X)^{-1}(\lambda\mathbf{1}+\nu), +\] +defined when $X$ has full column rank; the dual problem is $\max_{\lambda,\,\nu\geq\mathbf{0}}\,g(\lambda,\nu)$. The +feasible set $\{w:\mathbf{1}^\top w=1,\,w\geq\mathbf{0}\}$ is compact and $w=\tfrac{1}{n}\mathbf{1}$ is strictly +feasible, so Slater's condition holds and strong duality obtains: the primal and dual optima coincide and the duality +gap is zero. + +At the optimum, complementary slackness ($\nu_i w_i = 0$ for all $i$) partitions assets into two types. \emph{Active} +assets ($w_i>0$) have $\nu_i=0$; stationarity then gives $\frac{2}{T}(X^\top Xw)_i=\lambda$, so every active asset +contributes equally to marginal portfolio variance. \emph{Excluded} assets ($w_i=0$) have +$\nu_i = \frac{2}{T}(X^\top Xw)_i - \lambda \geq 0$: +their marginal variance is at least $\lambda$ and they would increase portfolio variance if held at a positive level. +The primal--dual loop in Section~\ref{sec:kkt} is a direct implementation of these conditions: the loop terminates +precisely when both primal feasibility and dual feasibility hold, which---together with stationarity from each inner +solve---is sufficient for global optimality. + +\subsection*{Problem variants.} + +The framework handles a range of standard portfolio construction problems directly. Each is expressed as +\begin{equation} + \min_{w}\;\|\tilde{X}w\|^2 - \rho\mu^\top w + \quad\text{subject to}\quad + \mathbf{1}^\top w = 1,\;\; w \geq 0, + \label{eq:general} +\end{equation} +through choices of $\tilde{X}$, $\rho$, and $\mu$ summarised in Table~\ref{tab:problems}. No change to the +algorithm is required; only the data passed to the solver changes. + +\begin{table}[ht] +\centering +\begin{tabular}{llll} +\toprule +Portfolio problem & $\tilde{X}$ & $\rho$ & $\mu$ \\ +\midrule +Minimum variance & $X$ & $0$ & --- \\ +Mean-variance (Markowitz) & $X$ & $>0$ & expected returns \\ +Minimum tracking error$^*$ & $X$ & $2$ & $X^\top(Xb)$ \\ +LW-regularised minimum variance & $[\sqrt{(1-\alpha)/T}\,X;\,\sqrt{\alpha}\,R]$ & $0$ & --- \\ +Factor-model covariance $FF^\top+D$ & $[F;\,D^{1/2}]$ & $0$ & --- \\ +Regime-blended / multi-period & $[\omega_1 X_1;\,\ldots;\,\omega_K X_K]$ & $0$ & --- \\ +\bottomrule +\end{tabular} +\caption{Portfolio construction problems solved by the same primal--dual loop. + $b$ is the benchmark weight vector; $F$ contains factor returns; $D$ is a diagonal specific-risk matrix; + $R$ is the $N\times N$ regularisation target matrix (e.g.\ $\sqrt{\bar\lambda}\,I$ for classical Ledoit-Wolf + shrinkage, where $\bar\lambda = \|X\|_F^2/(nT)$ is the mean eigenvalue of $\hat\Sigma$). + Stacking of $\tilde{X}$ is in the row direction. + $^*\min_w (w-b)^\top\Sigma(w-b) = \min_w w^\top\Sigma w - 2(\Sigma b)^\top w + \mathrm{const}$, so + $\rho\mu = 2\Sigma b = 2X^\top(Xb)$ (two matrix--vector products). + Since $\mu = \Sigma b$, the second SPD solve yields $v_2 = b$ exactly; tracking error adds no inner-solve cost.} +\label{tab:problems} +\end{table} + +\paragraph{What does not fit.} +The maximum Sharpe ratio (tangency) portfolio $\arg\max_w \mu^\top w / \sqrt{w^\top\Sigma w}$, subject to +$w \geq 0$, is a fractional programme and does not reduce to \eqref{eq:general} with a fixed $\rho$. It can be +recovered parametrically by sweeping $\rho$ over the efficient frontier and locating the tangency point, but this +requires multiple solves. Problems with additional linear inequality constraints---sector limits, turnover +bounds, factor-exposure limits---also lie outside the current template; a general QP solver (e.g.\ CVXPY with +Clarabel) is the appropriate tool for those. + +The one structural requirement is that the covariance admit a factorisation $\Sigma = \tilde{X}^\top\tilde{X}$ +with $\tilde{X}$ available directly or by cheap row-augmentation, so that the matrix-vector product +$v\mapsto\tilde{X}^\top(\tilde{X}v)$ remains $O(T'n)$ per iteration. Nonlinear shrinkage~\cite{ledoit2020}, +which applies a separate coefficient to each sample eigenvalue, does not admit such a factorisation and requires +a different preconditioning strategy. + +\section{Positive Definite Reduction and Primal-Dual Loop}\label{sec:kkt} + +Ignoring the inequality constraints for now, the Lagrangian for the equality-constrained problem is: +\[ +\mathcal{L}(w, \lambda) = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1) +\] +Setting the derivative to zero gives the stationarity condition +$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\mu$, where $\hat\Sigma = X^\top X/T$ (or $\Sigma_{\mathrm{LW}}$ with shrinkage). +For $\rho = 0$ this reduces to $\frac{2}{T}\hat\Sigma\, w = \lambda\mathbf{1}$, determining the solution up to a scalar. + +\paragraph{Positive definite reduction.} +When $\rho = 0$ (pure minimum variance), stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1}$, +so $w$ is a scalar multiple of $\hat\Sigma^{-1}\mathbf{1}$. +Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and writing $w = c\,v_1$, the budget constraint +$\mathbf{1}^\top w = 1$ fixes +\[ + c = \frac{1}{\mathbf{1}^\top v_1}, \qquad w = \frac{v_1}{\mathbf{1}^\top v_1}. +\] +When $\rho \neq 0$, stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1} + \rho\mu$, so +$w = \tfrac{\lambda}{2}\,\hat\Sigma^{-1}\mathbf{1} + \tfrac{\rho}{2}\,\hat\Sigma^{-1}\mu$. +Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and $v_2 = \hat\Sigma^{-1}\mu$ (two SPD solves), the budget +constraint determines the multiplier +\[ + \lambda = \frac{2 - \rho\,(\mathbf{1}^\top v_2)}{\mathbf{1}^\top v_1}, +\] +and the weights follow as $w = \tfrac{\lambda}{2}\,v_1 + \tfrac{\rho}{2}\,v_2$. +In both cases the $(n+1)\times(n+1)$ saddle-point system is never needed and $\lambda$ is +recovered analytically rather than as part of the solve. + +When $\alpha > 0$ and a regularisation target $R$ is provided, the regularised matrix +$\Sigma_{\mathrm{LW}} = \frac{1-\alpha}{T}\,X^\top X + \alpha\,R^\top R = (1-\alpha)\,\hat\Sigma + \alpha\,R^\top R$ +is symmetric positive definite and standard CG applies directly to $\Sigma_{\mathrm{LW}}\,v = \mathbf{1}$. +For classical Ledoit-Wolf shrinkage one sets $R = \sqrt{\bar\lambda}\,I$ with $\bar\lambda = \|X\|_F^2/(nT)$, +giving $\Sigma_{\mathrm{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\bar\lambda\,I$. +For completeness, the equivalent $(n+1)\times(n+1)$ saddle-point formulation +\[ +\begin{pmatrix} 2\Sigma_{\mathrm{LW}} & \mathbf{1} \\ \mathbf{1}^\top & 0 \end{pmatrix} +\begin{pmatrix} w \\ \lambda \end{pmatrix} += +\begin{pmatrix} \mathbf{0} \\ 1 \end{pmatrix} +\] +is symmetric but indefinite and requires MINRES \cite{paige1975} rather than CG; it is shown here for completeness. + +The long-only constraint $w \geq 0$ is enforced by a \textbf{primal-dual iterative procedure}. Each outer step solves +the equality-constrained (budget-only) $n_a \times n_a$ positive definite system over the current asset universe. In the +\emph{primal step}, any asset whose weight is negative is dropped. Once all active weights are non-negative, a +\emph{dual step} checks the KKT gradient condition for every excluded asset: for global optimality, every excluded asset +$i$ must satisfy +\[ + \nabla_i f(w) \;\geq\; \lambda, + \qquad + \nabla_i f(w) = 2\Bigl(\tfrac{1-\alpha}{T}(X^\top X w)_i + \alpha\,(R^\top R\,w)_i\Bigr) - \rho\mu_i, +\] +where $\lambda$ is the budget Lagrange multiplier and $R$ is the regularisation target matrix. An excluded asset that fails this +condition would decrease portfolio variance if added at a small positive level; it is re-added to the active set and the +loop repeats. The loop terminates when both primal and dual feasibility hold simultaneously; combined with stationarity +from the KKT solve, this is sufficient for global optimality. The feasible set is non-empty and compact, so the +constrained minimum exists. + +With Ledoit-Wolf shrinkage, the asset elimination loop converges rapidly in practice. Let $\bar\lambda = \|X\|_F^2/(nT)$ +denote the average eigenvalue of the sample covariance. The unconstrained minimiser satisfies $w^\star \propto +\Sigma_{\mathrm{LW}}^{-1}\mathbf{1}$. As $\alpha \to 1$, $\Sigma_{\mathrm{LW}} \to \bar\lambda I$ and $w^\star \to +\frac{1}{n}\mathbf{1}$, the equal-weight portfolio with all components strictly positive. By continuity there exists a +threshold $\alpha^\star \in (0,1)$ above which every component of the unconstrained solution is positive; for $\alpha > +\alpha^\star$ the first solve requires no asset to be dropped. The first-order expansion +\[ +\Sigma_{\mathrm{LW}}^{-1}\mathbf{1} \approx \tfrac{1}{\alpha\bar\lambda}\mathbf{1} - +\tfrac{1-\alpha}{\alpha^2\bar\lambda^2 T}X^\top X\mathbf{1}, +\] +valid for $\alpha$ close to one, shows that $\alpha^\star$ is governed by the spectral structure of $X^\top X$: assets +with large total covariance exposure are the last to turn positive as $\alpha$ increases. For the LW-optimal $\alpha$ +and dense equity returns, the threshold is typically exceeded and the first solve yields few negative components. Across +50 experiments with $n$ from 100 to 1000 and $T = 2n$, the elimination loop required 2--3 outer iterations in all cases; +runtime is dominated by the linear solves. + +\begin{algorithm}[ht] +\caption{Primal-dual outer loop}\label{alg:elim} +\begin{algorithmic}[1] +\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$;\; regularisation target $R$ (optional) +\Ensure Weight vector $w \in \mathbb{R}^n$;\; total inner iteration count $k$ +\State $\mathrm{active} \leftarrow \{1,\ldots,n\}$;\; $k \leftarrow 0$ +\Loop + \State $(w_{\mathrm{a}},\, k_{\mathrm{step}}) \leftarrow f(\mathrm{active})$;\; $k \leftarrow k + k_{\mathrm{step}}$ + \If{$\exists\, i \in \mathrm{active}$ with $w_{a,i} < -\varepsilon$} + \State $\mathrm{active} \leftarrow \mathrm{active} \setminus \{i : w_{a,i} < -\varepsilon\}$ + \hfill\textit{(primal step: drop violators)} + \Else + \State $w_i \leftarrow w_{a,i}$ for $i \in \mathrm{active}$;\; $w_i \leftarrow 0$ for $i \notin \mathrm{active}$ + \State $g_i \leftarrow 2\bigl(\tfrac{1-\alpha}{T}(X^\top X w)_i + \alpha\,(R^\top R\,w)_i\bigr) - \rho\mu_i$ + \State $\lambda \leftarrow \mathrm{mean}_{j \in \mathrm{active}}(g_j)$ + \State $\mathcal{V} \leftarrow \{i \notin \mathrm{active} : g_i < \lambda - \varepsilon\}$ + \If{$\mathcal{V} = \emptyset$} + \State \textbf{break} \hfill\textit{(KKT conditions satisfied; globally optimal)} + \EndIf + \State $\mathrm{active} \leftarrow \mathrm{active} \cup \mathcal{V}$ + \hfill\textit{(dual step: re-add violators)} + \EndIf +\EndLoop +\State \Return $w$, $k$ +\end{algorithmic} +\end{algorithm} + +Algorithm~\ref{alg:elim} accepts any inner solver $f$; Section~\ref{sec:solvers} presents four concrete choices: +matrix-free CG (the primary method), a direct dense factorisation, non-negative least squares (NNLS), and the CVXPY interior-point reference. + +\section{Computational Methods}\label{sec:solvers} + +\subsection{CG Method}\label{ssec:cg} + +The inner solver $f$ for Algorithm~\ref{alg:elim} is conjugate gradients applied to the $n_a \times n_a$ SPD system +derived in Section~\ref{sec:kkt}. The action of $\Sigma_{\mathrm{LW}}$ on a vector requires only two matrix-vector +products with $X_{\mathrm{a}}$; the matrix itself is never formed. + +\begin{algorithm}[ht] +\caption{CG inner step (inner solver $f$ for Algorithm~\ref{alg:elim})}\label{alg:cg} +\begin{algorithmic}[1] +\Require Active-asset index set $\mathrm{active}$;\; problem data;\; regularisation target $R$ (optional) +\Ensure Active-asset weights $w_{\mathrm{a}}$;\; CG iteration count $k$ +\Function{CgSolve}{active} +\State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $R_{\mathrm{a}} \leftarrow R[\mathrm{active},\mathrm{active}]$,\; $n_a \leftarrow + |\mathrm{active}|$ + \State Build matrix-free operator $\mathcal{S}$ of size $n_a \times n_a$: + \Statex\hspace{2em} $v \;\mapsto\; \tfrac{1-\alpha}{T}\,X_{\mathrm{a}}^\top(X_{\mathrm{a}}\,v) + \alpha\,R_{\mathrm{a}}^\top(R_{\mathrm{a}}\,v)$ + \hfill\textit{($\Sigma_{\mathrm{LW}}$ applied matrix-free)} + \State Solve $\mathcal{S}\,v = \mathbf{1}$ via CG + \State \Return $v\,/\,(\mathbf{1}^\top v)$,\; iteration count +\EndFunction +\end{algorithmic} +\end{algorithm} + +When an asset is dropped in the primal step, $\mathcal{S}$ is rebuilt for the reduced active set and CG restarts from +scratch on the smaller system. + +\paragraph{Convergence speed.} +The classical CG error bound \cite{greenbaum1997} states +\[ + \frac{\|e_k\|_{\Sigma_{\mathrm{LW}}}}{\|e_0\|_{\Sigma_{\mathrm{LW}}}} + \;\leq\; + 2\!\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{\!k}, + \qquad + \kappa = \kappa(\Sigma_{\mathrm{LW}}), +\] +where $\|e\|_{\Sigma} = (e^\top \Sigma e)^{1/2}$ is the energy norm and $\kappa$ is the spectral condition number +introduced in Section~\ref{sec:precond}. Shrinkage compresses $\kappa$ monotonically as $\alpha$ increases, directly +improving the asymptotic convergence rate. + +A second, sharper bound follows from the rank structure of $\Sigma_{\mathrm{LW}}$. +Because $X_{\mathrm{a}} \in \mathbb{R}^{T \times n_a}$ has rank at most $\min(T,n_a)$, the matrix +$\Sigma_{\mathrm{LW}} = \frac{1-\alpha}{T}X_{\mathrm{a}}^\top X_{\mathrm{a}} + \alpha R_{\mathrm{a}}^\top R_{\mathrm{a}}$ +has \emph{at most} $\min(T,n_a)+1$ distinct eigenvalues: exactly $n_a - \min(T,n_a)$ eigenvalues equal to +$\alpha\lambda_{\min}(R_{\mathrm{a}}^\top R_{\mathrm{a}})$ (the ridge floor) and up to $\min(T,n_a)$ distinct eigenvalues above it. +Since CG terminates in exact arithmetic in at most as many steps as there are distinct eigenvalues +\cite{greenbaum1997}, convergence requires at most $\min(T,n_a)+1$ iterations. +In the underdetermined regime $T < n_a$ this yields a bound of $T+1$, independent of the active-set size~$n_a$; +adding assets to the portfolio does not increase the iteration count. +This regime arises when the history window is shorter than the universe (a common scenario in, e.g., +high-frequency or short-window portfolio rebalancing), or within the primal-dual loop once enough assets have been +excluded that $n_a$ exceeds $T$. Shrinkage reinforces the rank-based bound by clustering the $\min(T,n_a)$ +non-ridge eigenvalues, causing super-linear acceleration in practice \cite{simoncini2013}. +The iteration counts in Table~\ref{tab:sp500} illustrate the condition-number effect: +on the S\&P~500 problem ($n=495$, $T=1192 > n$) shrinkage reduces CG iterations from 744 to 82, consistent +with $\kappa(\Sigma_{\mathrm{LW}})$ decreasing monotonically as $\alpha$ increases (Section~\ref{sec:precond}). + +\subsection{CVXPY Reference Implementation} + +Before developing the Krylov solvers we establish a reference solution using CVXPY \cite{diamond2016}, a domain-specific +language for convex optimisation. The minimum variance problem maps directly onto a least-squares objective, which CVXPY +dispatches to a general-purpose interior-point solver. + +\begin{algorithm}[ht] +\caption{CVXPY reference solver}\label{alg:cvxpy} +\begin{algorithmic}[1] +\Require Returns matrix $X \in \mathbb{R}^{T \times n}$;\; shrinkage $\alpha$;\; return tilt $\rho\mu$;\; target $R$ +\Ensure Weight vector $w$;\; interior-point iteration count $k$ +\State Declare $w \in \mathbb{R}^n$ as a CVXPY variable +\State Minimise $\tfrac{1-\alpha}{T}\|Xw\|^2 + \alpha\|Rw\|^2 - \rho\mu^\top w$ + \Statex\hspace{1.5em} subject to $\mathbf{1}^\top w = 1$,\; $w \geq \mathbf{0}$ +\State Dispatch to Clarabel interior-point solver \cite{clarabel2024} +\State \Return $w$,\; $k$ +\end{algorithmic} +\end{algorithm} + +The formulation is transparent and correct, and CVXPY handles the long-only constraint natively. We use the Clarabel +interior-point solver \cite{clarabel2024}, which dispatches the problem as a second-order cone programme. Its cost is +substantial: interior-point methods require $O(n^3)$ work per iteration, and the overhead of problem construction and +solver communication adds further latency. On the S\&P~500 problem ($n = 495$) this amounts to roughly 1.6 seconds +(Table~\ref{tab:sp500}), making CVXPY unsuitable for production use but ideal as a ground-truth reference. + +All three methods in this paper (CVXPY, KKT direct, and CG) solve the same optimisation problem and produce identical +portfolio weights up to numerical tolerance. The comparison is purely computational: the question is not which method +finds a better portfolio, but which finds the same portfolio fastest. + +The speedup figures in Table~\ref{tab:sp500} should be read with this in mind. CVXPY and +Clarabel are general-purpose tools: the same code handles additional convex constraints --- sector limits, turnover +bounds, factor exposure constraints --- without modification. The CG method exploits the specific structure of the pure +long-only minimum variance problem and would require rework to accommodate constraints beyond the budget and +non-negativity requirements. The speedup therefore conflates two effects: genuine algorithmic efficiency from exploiting +problem structure, and the overhead of a framework designed for a far broader problem class. For the standard long-only +constraint the full speedup is available; practitioners requiring richer constraint sets face a narrower tradeoff and +may find a specialised QP solver a more appropriate baseline. + +\subsection{Direct Solve (Baseline)} + +For small $n$, \texttt{numpy.linalg.solve} applied to the $n_a \times n_a$ covariance matrix is fast and exact. It +requires forming $X_{\mathrm{a}}^\top X_{\mathrm{a}}$ (an $O(n_a^2 T)$ operation), but for $n \ll T$ this is not the +bottleneck. + +\begin{algorithm}[ht] +\caption{Direct solver}\label{alg:kkt} +\begin{algorithmic}[1] +\Require Problem data;\; regularisation target $R$ (optional) +\Ensure Weight vector $w$;\; outer elimination step count $s$ +\Function{DirectSolve}{active} +\State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $R_{\mathrm{a}} \leftarrow R[\mathrm{active},\mathrm{active}]$,\; $n_a \leftarrow + |\mathrm{active}|$ + \State Assemble $\Sigma_{\mathrm{a}} \leftarrow \tfrac{1-\alpha}{T}\,X_{\mathrm{a}}^\top X_{\mathrm{a}} + \alpha\,R_{\mathrm{a}}^\top R_{\mathrm{a}}$ + \hfill\textit{(SPD, $n_a\times n_a$)} + \State Solve $\Sigma_{\mathrm{a}}\, v = \mathbf{1}$ by dense factorisation + \State \Return $v\,/\,(\mathbf{1}^\top v)$,\; $1$ \hfill\textit{(per-step count; no inner iterations)} +\EndFunction +\State $(w,\, s) \leftarrow \textsc{EliminationLoop}(\textsc{DirectSolve})$ + \hfill\textit{(Algorithm~\ref{alg:elim})} +\State \Return $w$, $s$ +\end{algorithmic} +\end{algorithm} + + + +\subsection{NNLS Solver}\label{ssec:nnls} + +An alternative approach bypasses the primal-dual loop entirely by reformulating the long-only constraint as a +non-negative least squares (NNLS) problem. Starting from the objective $\min_w \frac{1-\alpha}{T}\|Xw\|^2 + +\alpha\|Rw\|^2$, the budget constraint $\mathbf{1}^\top w = 1$ is incorporated via a large-penalty row, yielding the +augmented system: +\[ + \min_{w \geq 0}\; \left\|\begin{pmatrix} \sqrt{(1-\alpha)/T}\,X \\ \sqrt{\alpha}\,R \\ M\,\mathbf{1}^\top \end{pmatrix} + w - \begin{pmatrix} \mathbf{0} \\ \mathbf{0} \\ M \end{pmatrix}\right\|^2, +\] +where $M = \|X\|_F \cdot T$ is a penalty large enough to enforce $\mathbf{1}^\top w \approx 1$ to machine precision. +The non-negativity constraint $w \geq 0$ is handled natively by the Lawson-Hanson active-set algorithm +\cite{lawson1974}, which solves NNLS by iterating over the free-variable set rather than forming an explicit KKT +system. + +This formulation has two practical advantages: it is single-shot (no outer primal-dual loop), and the budget constraint +requires no post-processing. However, NNLS does not benefit from the matrix-free structure of CG: the Lawson-Hanson +algorithm implicitly forms normal equations $A^\top A$ where $A$ is the augmented matrix, incurring the same $O(n^2 T)$ +cost as the direct KKT solve. With Ledoit-Wolf shrinkage, the augmented matrix grows from $T \times n$ to $(T+n) \times +n$, making NNLS paradoxically slower with shrinkage than without, an inversion of the pattern seen for KKT and CG. +Table~\ref{tab:sp500} confirms this: on the S\&P~500 problem NNLS takes $0.038$\,s without +shrinkage but $0.086$\,s with it, while CG moves in the opposite direction. + +\subsection{Proximal Gradient}\label{ssec:proximal} + +An alternative to the active-set loop is to enforce the simplex constraint directly at every +iteration via Euclidean projection. The minimum variance objective with Ledoit-Wolf shrinkage +can be written as +\[ + \min_{w \in \Delta_n}\;\tfrac{1}{2}\|\tilde{X}w\|^2, + \qquad + \Delta_n = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 1,\; w \geq \mathbf{0}\}, +\] +where $\tilde{X} = [\sqrt{(1-\alpha)/T}\,X;\,\sqrt{\alpha}\,R]$ as before and the probability +simplex $\Delta_n$ captures both budget and long-only constraints simultaneously. The gradient +of the objective is $\nabla f(w) = \Sigma_{\mathrm{LW}}\,w$ with +$\Sigma_{\mathrm{LW}} = \tilde{X}^\top\tilde{X}$; the Lipschitz constant of the gradient is +$L = \lambda_{\max}(\Sigma_{\mathrm{LW}})$. The projected gradient iteration is +\begin{equation}\label{eq:proxgrad} + w_{k+1} = \Pi_{\Delta_n}\!\left(w_k - \tau\,\Sigma_{\mathrm{LW}}\,w_k\right), + \qquad \tau = \frac{1}{2L}, +\end{equation} +where $\Pi_{\Delta_n}$ denotes Euclidean projection onto the simplex. + +\paragraph{Simplex projection.} +Duchi et al.\ \cite{duchi2008} give an $O(n\log n)$ closed-form algorithm for $\Pi_{\Delta_n}$: +sort $v$ in decreasing order to obtain $\mu_1 \geq \cdots \geq \mu_n$, find the largest index +$\rho$ such that $\mu_\rho > \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, set +$\theta = \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, and return +$[\Pi_{\Delta_n}(v)]_i = \max(v_i - \theta,\,0)$. The projection requires a single sort and a +linear scan; it is the inner loop of Algorithm~\ref{alg:proximal}. + +\begin{algorithm}[ht] +\caption{Proximal gradient solver}\label{alg:proximal} +\begin{algorithmic}[1] +\Require Stacked return matrix $\tilde{X} \in \mathbb{R}^{(T+n_R) \times n}$;\; + tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ +\Ensure Weight vector $w \in \Delta_n$ +\State Assemble $\Sigma_{\mathrm{LW}} \leftarrow \tilde{X}^\top\tilde{X}$ + \hfill\textit{($n\times n$, assembled once)} +\State $L \leftarrow \lambda_{\max}(\Sigma_{\mathrm{LW}})$;\quad $\tau \leftarrow (2L)^{-1}$ +\State Initialise $w_0 \in \mathbb{R}^n$ randomly;\; $k \leftarrow 0$ +\Loop + \State $w_{k+1} \leftarrow \Pi_{\Delta_n}(w_k - \tau\,\Sigma_{\mathrm{LW}}\,w_k)$ + \hfill\textit{(gradient step + simplex projection)} + \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$ \textbf{or} $k \geq k_{\max}$} + \textbf{break} + \EndIf + \State $k \leftarrow k + 1$ +\EndLoop +\State \Return $w_{k+1}$ +\end{algorithmic} +\end{algorithm} + +\paragraph{Cost per iteration.} +The matrix $\Sigma_{\mathrm{LW}}$ is assembled once at $O(n^2(T+n_R))$ cost (the same setup as +the direct KKT solve) and cached; the largest eigenvalue $L$ is computed alongside it. Each +subsequent iteration costs $O(n^2)$ for the matrix-vector product and $O(n\log n)$ for the +simplex projection, dominated by the $O(n^2)$ term. + +\paragraph{Convergence.} +Since $\Sigma_{\mathrm{LW}}$ is SPD, the objective is $\lambda_{\min}(\Sigma_{\mathrm{LW}})$-strongly +convex on $\Delta_n$. Projected gradient descent with step $\tau \leq 1/L$ converges linearly: +\[ + \|w_k - w^\star\|^2 + \;\leq\; + \left(1 - \frac{1}{\kappa(\Sigma_{\mathrm{LW}})}\right)^{\!k} + \|w_0 - w^\star\|^2. +\] +The iteration count therefore scales as $O(\kappa(\Sigma_{\mathrm{LW}}))$, compared with +$O(\sqrt{\kappa})$ for CG. On ill-conditioned problems (e.g.\ the unshrunk S\&P~500 system +with $\kappa \approx 10^5$) this gap is decisive. Ledoit-Wolf shrinkage compresses +$\kappa(\Sigma_{\mathrm{LW}})$ monotonically as $\alpha$ increases and benefits both methods, +but it narrows the ratio of iteration counts only slowly. + +\paragraph{Comparison with other solvers.} +The proximal solver occupies a niche between NNLS and the active-set methods. Like NNLS it +handles the long-only and budget constraints in a single stage with no outer primal-dual loop. +Unlike NNLS, it does not form the normal equations at each step: NNLS incurs an implicit +$O(n^2)$ cost per Lawson-Hanson step \emph{and} its augmented matrix grows from $T \times n$ to +$(T+n)\times n$ when shrinkage is applied, paradoxically making NNLS slower with shrinkage. +The proximal solver's augmented matrix is fixed at $(T+n_R)\times n$ regardless of the +shrinkage path; on the S\&P~500 problem this gives $0.039$\,s with LW shrinkage versus +$0.11$\,s for NNLS. + +At large $n$ the proximal method avoids the $O(n^3)$ factorisation of KKT direct. The scaling +experiments (Figure~\ref{fig:loglog}) show it sitting between CG and KKT: at $n=2000$ the +proximal solver takes $0.35$\,s, versus $0.27$\,s for CG and $0.51$\,s for KKT direct. CG +retains the edge throughout because its $O(\sqrt{\kappa})$ iteration count and $O(nT)$ per-step +cost (exploiting the matrix-free structure) compound favourably, whereas the proximal solver +pays $O(n^2)$ per step regardless of the matrix-free potential. + +The proximal gradient approach is most attractive when the simplex-projection structure is +exploited further---for instance in the $\ell_1$-ball variant of the long-only constraint +studied in \cite{duchi2008}---or when one prefers a conceptually simple, loop-free +implementation at the cost of slower per-problem convergence. + +\section{Ledoit-Wolf Shrinkage} +\label{sec:precond} + +The sample covariance matrix $\frac{1}{T}X^\top X$ is a poor estimator when $n$ is large relative to $T$: its smallest +eigenvalues are severely downward-biased, and the eigenvalue spread (the ratio of largest to smallest) far exceeds that +of the true covariance. Ledoit and Wolf \cite{ledoit2004} propose replacing the sample covariance with a convex +combination of the sample estimate and a scaled identity: +\[ +\Sigma_{\text{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\cdot\mathrm{target}, \quad \bar\lambda = \frac{\|X\|_F^2}{nT} +\] +where $\bar\lambda$ is the average eigenvalue of the sample covariance and the classical choice is +$\mathrm{target} = \bar\lambda I$. The implementation uses a general $N\times N$ regularisation target matrix $R$ +(satisfying $\mathrm{target} = R^\top R$), so that +$\Sigma_{\text{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\,R^\top R$. +Classical Ledoit-Wolf shrinkage is recovered by setting $R = \sqrt{\bar\lambda}\,I$; for i.i.d.\ $\mathcal{N}(0,1)$ +synthetic returns $\bar\lambda \approx 1$ so $R = I$ is equivalent. +The shrinkage intensity $\alpha \in (0,1)$ is +chosen analytically to minimise the expected Frobenius distance to the true covariance, requiring no cross-validation. +The effect on the spectrum is direct: the target term lifts every eigenvalue, bounding the smallest away from zero and +compressing the overall spread. The condition number satisfies: +\[ +\kappa(\Sigma_{\text{LW}}) = \frac{\lambda_{\max}(\Sigma_{\text{LW}})}{\lambda_{\min}(\Sigma_{\text{LW}})} \leq +\frac{(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)}{\alpha\lambda_{\min}(R^\top R)} +\] +since $\lambda_{\min}(\Sigma_{\text{LW}}) \geq \alpha\lambda_{\min}(R^\top R)$ and, by Weyl's inequality applied to +$\Sigma_{\text{LW}} = (1-\alpha)\hat\Sigma + \alpha R^\top R$, $\lambda_{\max}(\Sigma_{\text{LW}}) \leq +(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)$, +where $\lambda_{\max}(\hat\Sigma) = \lambda_{\max}(X^\top X/T)$ is +the largest sample eigenvalue. As $\alpha \to 1$ the bound approaches $\lambda_{\max}(R^\top R)/\lambda_{\min}(R^\top R)$, +which equals $1$ for the classical choice $R = \sqrt{\bar\lambda}\,I$, correctly reflecting that full shrinkage +yields a scalar multiple of the identity. For fixed data, the condition number decreases monotonically as $\alpha$ +increases, with the rate governed by the spectral ratio $\lambda_{\max}(\hat\Sigma)/\lambda_{\min}(R^\top R)$. For CG applied to +$\Sigma_{\text{LW}}\,v = \mathbf{1}$, the convergence rate is governed by $\kappa(\Sigma_{\text{LW}})$; compressing the +spectrum directly reduces iteration counts. Subsequent work by the same authors provides an analytical nonlinear +shrinkage formula \cite{ledoit2020} that applies a different coefficient to each sample eigenvalue, achieving a smaller +Frobenius loss when $n/T$ is not negligible. Nonlinear shrinkage cannot be expressed as a simple augmented matrix +$\tilde{X}$, so incorporating it into the matrix-free framework would require a separate Krylov-compatible +preconditioner rather than a modified return matrix. + +In the implementation, shrinkage is applied by augmenting the return matrix rather than forming $\Sigma_{\text{LW}}$ +explicitly. Setting $c = 1 - \alpha$, the stacked matrix $\tilde{X} += [\sqrt{c/T}\,X;\,\sqrt{\alpha}\,R]$ satisfies $\tilde{X}^\top \tilde{X} = \frac{c}{T}\,X^\top X + \alpha\,R^\top R += c\,\hat\Sigma + \alpha\,R^\top R = \Sigma_{\text{LW}}$, +so any of the solvers above can be applied to $\tilde{X}$ and produce exactly the LW-shrinkage portfolio weights. +For $R = \sqrt{\bar\lambda}\,I$ this gives classical Ledoit-Wolf: $\Sigma_{\text{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\bar\lambda\,I$. + +From the perspective of numerical linear algebra, compressing the eigenvalue spectrum of the system matrix is precisely +what a preconditioner does. Ledoit-Wolf shrinkage therefore plays a dual role: it reduces statistical estimation error +and, as a side effect, accelerates iterative solvers by clustering the eigenvalues. The two motivations are independent +but arrive at the same modified matrix. + +The proportional identity shrinkage target $\alpha\bar\lambda I$ is the simplest choice, but the preconditioning +literature suggests sharper alternatives. A factor-model preconditioner, replacing $\alpha\bar\lambda I$ with a +low-rank-plus-diagonal approximation $BB^\top + D$, would cluster eigenvalues more tightly at the cost of a slightly +more expensive application. Incomplete Cholesky factorisation of the shrunk matrix provides a general-purpose option. +For rolling-window problems where the covariance changes slowly between periods, a preconditioner built from the +previous period's factorisation could be reused with minimal update cost. None of these require forming the covariance +matrix explicitly and each is compatible with the matrix-free framework described above. + +\section{Numerical Results}\label{sec:results} + +All experiments were run on an Apple M4 Pro (14-core CPU, 48\,GB RAM) under Python~3.12, NumPy~2.4, SciPy~1.17, +CVXPY~1.8.2, and Clarabel~0.11.1. The iterative solvers use SciPy's default relative residual tolerance of $10^{-5}$, +which was reached in all reported runs; reported times are the minimum of three repeated solves and cover only the +solve itself --- construction of the shrinkage target $\hat\sigma^2 I$ is excluded, as it requires a single pass over +$X$ and is negligible relative to any solver. All implementations are available in the companion package +\cite{fast_minimum_variance}. + + +\subsection*{S\&P~500 benchmark} + +The dataset consists of daily percentage returns for 495 S\&P~500 constituents over the period July~2021--May~2026 +(1192 trading days, $T/n \approx 2.4$), sourced from Yahoo Finance via the \texttt{yfinance} Python package. +Tickers with more than 5\% missing observations are excluded; short gaps are forward-filled. The return +matrix $X$ is formed by demeaning the returns column-wise. + +\paragraph{Spectrum of $\hat\Sigma = \tfrac{1}{T}X^\top X$.} +The eigenvalue structure of the sample covariance is qualitatively different from that of a random Gaussian matrix and +is the reason the S\&P~500 problem is a meaningful test case for the preconditioning argument. For a random $T \times n$ +matrix with i.i.d.\ $\mathcal{N}(0,\sigma^2)$ entries, the Marchenko-Pastur law predicts that the bulk of the spectrum +lies in the interval $[\sigma^2(1-\sqrt{n/T})^2,\,\sigma^2(1+\sqrt{n/T})^2]$, giving a mild condition number that +grows modestly with $n/T$. The equity spectrum is nothing like this. + +The largest eigenvalue is $\lambda_1 \approx 0.067$, the market factor, which alone accounts for $\approx 30\%$ of +total portfolio variance. To apply the Marchenko-Pastur law we estimate $\sigma^2$ as the average squared entry of $X$, +\[ + \hat\sigma^2 = \frac{\|X\|_F^2}{nT} \approx 0.000457, +\] +the natural entry-level variance of a daily percentage return. The MP upper edge $\hat\sigma^2(1+\sqrt{n/T})^2 \approx 0.00123$ +then identifies 22 eigenvalues as statistically significant factors; the remaining 473 are consistent with noise yet +still span a wide range down to $\lambda_{\min} \approx 6 \times 10^{-7}$. The resulting condition number is +\[ + \kappa(\hat\Sigma) = \frac{\lambda_{\max}}{\lambda_{\min}} \approx 109{,}000, +\] +far larger than the $O(10)$--$O(100)$ range typical of random matrices at this aspect ratio. The first ten +eigenvalues account for half the total variance, while the bottom half of the spectrum contributes less than $3\%$. +This extreme imbalance---a handful of dominant factors and a long tail of near-zero noise eigenvalues---is precisely +what makes CG slow without preconditioning (744 iterations) and what makes shrinkage decisive: lifting +$\lambda_{\min}$ by even a moderate $\alpha$ collapses the condition number and cuts the iteration count to 82. + +Table~\ref{tab:sp500} reports timings and iteration counts for all solvers on this dataset. + +\begin{table}[ht] +\centering +\begin{tabular}{lccc} +\toprule +Method & Time (s) & Iterations & Speedup \\ +\midrule +\multicolumn{4}{l}{\textit{Without Ledoit-Wolf shrinkage}} \\[2pt] +cvxpy (Clarabel) & 1.778 & 11 & 1.0x \\ +Clarabel (direct API) & 0.071 & 11 & 25.1x \\ +KKT direct & 0.0056 & 22 & 315.3x \\ +CG (SPD) & 0.0120 & 744 & 148.1x \\ +NNLS (Lawson--Hanson) & 0.0368 & -- & 48.4x \\ +\midrule +\multicolumn{4}{l}{\textit{With Ledoit-Wolf shrinkage ($\alpha = 0.5$)}} \\[2pt] +cvxpy (Clarabel) & 2.482 & 16 & 1.0x \\ +Clarabel (direct API) & 0.092 & 16 & 27.0x \\ +KKT direct & 0.0062 & 6 & 399.0x \\ +CG (SPD) & 0.0042 & 82 & 594.4x \\ +NNLS (Lawson--Hanson) & 0.1113 & -- & 22.3x \\ +\bottomrule +\end{tabular} +\caption{S\&P~500 universe: 495 assets, 1192 trading days (Jul~2021--May~2026). Speedup relative to \texttt{cvxpy} +within each panel. ``Clarabel (direct API)'' calls the same interior-point solver without CVXPY's overhead; $>95\%$ of +\texttt{cvxpy}'s time is Python interface work. NNLS is slower with shrinkage because the augmented matrix grows from +$T \times n$ to $(T+n) \times n$.} +\label{tab:sp500} +\end{table} + +The real-data results reveal two effects. First, without shrinkage the equity system is hard to solve iteratively: +CG requires 744 iterations and is slower than KKT direct (0.0120\,s vs.\ 0.0056\,s). The strong factor structure of +equity returns concentrates variance along a few eigendirections, inflating the condition number and offsetting the +matrix-free advantage of CG. + +Second, Ledoit-Wolf shrinkage benefits both iterative and direct solvers: it cuts CG iterations from 744 to~82 and +reduces active-set outer steps from~22 to~6, since a lower condition number means fewer assets end up with negative +unconstrained weights and the primal-dual loop converges faster. At $n=495$, KKT direct (0.0062\,s, $399\times$ over +CVXPY) and CG (0.0042\,s, $594\times$) are both well under a millisecond and within a factor of~1.5 of each other. +For portfolios of this size KKT direct is the practical recommendation: it requires no convergence tolerance and is +faster without shrinkage. CG's advantage lies at larger $n$, where avoiding explicit assembly of +$\tilde{X}^\top\tilde{X}$ reduces per-step cost from $O(n^2)$ to $O(Tn)$ (Figure~\ref{fig:scaling}). +The ``Clarabel (direct API)'' row confirms the speedup is not merely framework overhead: bypassing CVXPY reduces +Clarabel's time from 2.5\,s to 0.092\,s, yet KKT direct ($0.006$\,s) is still $15\times$ faster. These results are +specific to this five-year window; iteration counts depend on the factor structure of equity returns and may differ +in other market regimes. + +\subsection*{Scaling with problem size} + +Figure~\ref{fig:scaling} uses synthetic i.i.d.\ Gaussian data to isolate the computational scaling of each solver +across a range of problem sizes and shrinkage levels; the spectral properties of the data are irrelevant here since the +point is to measure cost growth, not iteration counts. The left panel shows wall-clock time as a function of $n$ (with +$T = 2n$ and LW shrinkage throughout). The KKT direct solve +scales as $O(n^3)$, dominated by the $n \times n$ linear system. Each Krylov iteration requires $O(nT) = O(n^2)$ work +for the two matrix-vector products with $X$; with the spectrum compressed by shrinkage the iteration count grows slowly +with $n$, giving an empirical scaling close to $O(n^2)$ and a gap that widens visibly beyond $n \approx 300$. + +The right panel uses the rank-deficient setting $T = 250 < n = 500$, in which $X^\top X$ has at most $T$ non-zero +eigenvalues and the unshrunk sample covariance is singular. This regime is practically relevant: asset universes with +more instruments than trading days arise when securities with limited common history are pooled, or when a large +cross-section is combined with a short lookback window. Without shrinkage the KKT system is singular and the iterative +solvers do not converge; every point in the panel has $\alpha > 0$. The preconditioning effect of shrinkage is strongest +precisely here: iteration counts fall steeply as $\alpha$ increases from zero, reaching the practical operating point at +moderate $\alpha$ values well below unity. + +\begin{figure}[ht] +\centering +\includegraphics[width=\linewidth]{graphs/minvar_scaling.pdf} +\caption{Left: wall-clock time vs.\ number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied +throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). Right: CG iteration counts vs.\ shrinkage +intensity $\alpha$ at fixed $n=500$, $T=250$ (rank-deficient baseline). Larger $\alpha$ compresses the eigenvalue +spectrum and reduces the condition number $\kappa(\Sigma_{\mathrm{LW}})$, directly reducing CG iterations.} +\label{fig:scaling} +\end{figure} + +\begin{figure}[ht] +\centering +\includegraphics[width=0.65\linewidth]{graphs/minvar_loglog.pdf} +\caption{Wall-clock time vs.\ number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied +throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). KKT direct scales as $O(n^3)$; matrix-free +CG scales empirically as $O(n^2)$ due to the iteration count growing slowly with the compressed spectrum; NNLS also +scales as $O(n^3)$ (Lawson--Hanson normal equations). Reference slopes are anchored to the KKT curve at $n = 500$.} +\label{fig:loglog} +\end{figure} + +\section{Conclusion} + +The long-only minimum variance portfolio problem, formulated directly in terms of the returns matrix, +reduces to solving the $n\times n$ symmetric positive definite system $\Sigma_{\mathrm{LW}}\, v = \mathbf{1}$ via +conjugate gradients, followed by a single rescaling $w = v/(\mathbf{1}^\top v)$. +The covariance matrix is never formed; all operations are matrix-vector products with $X$. + +The empirical results on 495 S\&P~500 constituents over five years confirm the practical value of this approach. +For portfolios up to $n \approx 1000$, KKT direct is already 300--400$\times$ faster than CVXPY across both +shrinkage regimes and requires no convergence tolerance: it is the practical recommendation. Without shrinkage, the +matrix-free advantage of CG is offset by high iteration counts (744 on S\&P~500 equity data) driven by the strong +factor structure of equity returns. Ledoit-Wolf shrinkage \cite{ledoit2004} benefits both solvers: it compresses +the eigenvalue spectrum, cuts CG iterations to~82, and reduces active-set outer steps from~22 to~6, since a lower +condition number shrinks the unconstrained minimum-variance portfolio toward equal weights and fewer assets receive +negative weights in the primal step. At $n=495$ CG (0.0042\,s) edges KKT direct (0.0062\,s) by a factor of~1.5. +CG's advantage becomes decisive at large $n$: the scaling experiments show CG is roughly $2\times$ faster than KKT +at $n=2000$, a gap that widens further as $n$ grows. + +The shrinkage estimator of Ledoit and Wolf and the preconditioning theory of Golub and Van Loan \cite{golub2013} address +different problems but act on the same object: the eigenvalue spectrum of the system matrix. This connection, which the +two literatures do not usually make explicit, points toward richer preconditioners (factor-model or low-rank structured) +as a productive direction for future work. + +The SPD structure extends naturally beyond the minimum variance setting. Adding a return term $\rho\mu^\top w$ to the +objective requires solving two systems ($\Sigma_{\mathrm{LW}}\, v_1 = \mathbf{1}$ and $\Sigma_{\mathrm{LW}}\, v_2 = \mu$) and recovering the budget +multiplier analytically; the full Markowitz mean-variance problem is handled in a companion paper. + + + +\bibliographystyle{siam/siamplain} +\bibliography{bib/refs} + +\end{document} From 47b6ca6d2afbdda9695880d11127770e3d5d8924 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:40:34 +0400 Subject: [PATCH 08/39] docs: split sec:kkt into Positive-Definite Reduction and Primal-Dual Loop Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 684 ++++++++++++++++++++++++----------------- 1 file changed, 406 insertions(+), 278 deletions(-) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index a10769c..6f3ad6a 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -6,6 +6,14 @@ \usepackage{xcolor} \usepackage{graphicx} \usepackage{algpseudocode} +\usepackage{authblk} + +\newcommand{\tcr}{\textcolor{red}} +\newcommand{\tcm}{\textcolor{magenta}} +\newcommand{\tcg}{\textcolor{green}} +\newcommand{\tcb}{\textcolor{blue}} +\newcommand{\norm}[1]{\lVert #1 \rVert} + \newsiamremark{remark}{Remark} \crefname{remark}{Remark}{Remarks} @@ -26,15 +34,15 @@ \title{Krylov Methods for the Minimum Variance Portfolio} -\author{Thomas Schmelzer% - \thanks{Jebel Quant Research (\email{thomas@jqr.ae})}% - \and - Michael Wolf% - \thanks{Department of Economics, University of Zurich.}% - \and - Martin Stoll% - \thanks{Faculty of Mathematics, TU~Chemnitz.}% -} +\author[1]{Thomas Schmelzer} +\author[2]{Martin Stoll} +\author[3,4]{Michael Wolf} + +\affil[1]{\small Jebel Quant Research, Abu Dhabi} +\affil[2]{\small Faculty of Mathematics, TU Chemnitz} +\affil[3]{\small Department of Economics, University of Zurich} +\affil[4]{\small ADIA Lab, Abu Dhabi} + \headers{Krylov Methods for the Minimum Variance Portfolio}{T.~Schmelzer, M.~Wolf, and M.~Stoll} @@ -44,19 +52,19 @@ \begin{abstract} The long-only minimum variance portfolio is routinely solved by assembling a sample covariance -matrix and dispatching it to a general-purpose quadratic programme. Both steps are unnecessary +matrix and dispatching it to a general-purpose quadratic program. Both steps are unnecessary and expensive. Conjugate gradients applied matrix-free to the returns matrix $X$ solve the problem in $O(nT)$ work per iteration without ever forming the $n\times n$ covariance; a primal-dual active-set loop enforces non-negativity with negligible overhead. The covariance matrix is not a primitive -of the problem --- it is a computational detour. +of the problem---it is a computational detour. Ledoit-Wolf shrinkage, already standard practice for statistical reasons, turns out to be identical to spectral preconditioning of the linear system: the same ridge term that shrinks -sample eigenvalues toward their mean also compresses the condition number and cuts CG iteration -counts dramatically. Statistical regularisation and numerical preconditioning are the same -operation applied to the same matrix --- a connection that does not appear to have been made +sample eigenvalues toward their mean also compresses the condition number and reduces CG iteration +counts dramatically. Statistical regularization and numerical preconditioning are the same +operation applied to the same matrix, a connection that does not appear to have been made explicit in the existing literature. Numerical experiments on S\&P~500 equity data and synthetic problems of growing size show @@ -68,51 +76,91 @@ \section{Introduction} -The minimum variance portfolio allocates capital across $n$ assets so as to minimise return variance subject only to -full investment and non-negativity. Since \cite{markowitz1952} it has been standard practice to estimate a sample -covariance matrix $\hat\Sigma = X^\top X / T$ from a $T \times n$ return matrix $X$ and pass it to a general-purpose -quadratic programming solver. This paper argues that both steps are unnecessary and expensive. +The long-only minimum variance portfolio allocates capital across $n$ assets to minimize portfolio return variance subject to two constraints: the weights must sum to one (the portfolio is fully invested) and must be non-negative. The +only input required in theory is the $n \times n$ +covariance matrix $\Sigma$ of the $n \times 1$ +vector of asset returns. In practice, $\Sigma$ is unknown and must be estimated from historical data. To this end, one uses a $T\times n$ matrix of demeaned returns $X$, where $T$ denotes the length of the estimation window (or the sample size). The classical estimator of $\Sigma$ is the sample covariance matrix $\hat \Sigma = X^\top X/T$. + +%Since Markowitz \cite{markowitz1952} it has been standard practice %\tcr{Did Markowitz already discuss estimation of $\Sigma$?} \tcb{He only had a $3 \times 3$ matrix in there} to estimate $\Sigma$ by the sample +%covariance matrix $\hat\Sigma = X^\top X / T$ from a $T \times n$ demeaned return matrix $X$ and pass it to a general-purpose +%quadratic programming solver. \tcb{We introduce a method bypassing the explicit computation and storage of the sample covariance matrix $\Sigma$. %It is sufficient to compute the product of $\Sigma v$ which boils down to two matrix-vector products $X^\top X v$. +%} +%This paper argues that both steps are unnecessary and expensive. +%\tcr{This is a bit confusing. What is meant by the first step? Estimating $\Sigma$? That is certainly necessary. Estimating $\Sigma$ by the sample covariance matrix?} -That the minimum-variance solution satisfies $w \propto \Sigma^{-1}\mathbf{1}$ is a classical observation: it follows -directly from the KKT stationarity condition $2\Sigma w = \lambda\mathbf{1}$ and was made explicit by -\cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the single rescaling -$w = v/(\mathbf{1}^\top v)$ where $v = \Sigma^{-1}\mathbf{1}$. What has \emph{not} been done is to exploit this -structure computationally: the standard pipeline still forms $\Sigma = X^\top X/T$ explicitly before solving. +\tcb{ +The Lagrangian for the problem +\[ +\min_w\; w^\top \hat \Sigma w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, +\] +where $\mathbf{1}$ denotes a conformable vector of ones, is +\[ +\mathcal{L}(w,\lambda) = w^\top \hat \Sigma w - \lambda(\mathbf{1}^\top w - 1), +\] +whose stationarity condition $2 \hat \Sigma w = \lambda\mathbf{1}$ immediately gives $w \propto \hat \Sigma^{-1}\mathbf{1}$; this was made explicit by \cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the +single rescaling +\[ +w = v/(\mathbf{1}^\top v), \qquad v = \hat \Sigma^{-1}\mathbf{1}. +\] +What has \emph{not} been done is to exploit this structure computationally: the standard pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. +} -This paper makes two contributions. First, we apply conjugate gradients (CG) to $\Sigma v = \mathbf{1}$ -\emph{matrix-free}: $\Sigma = X^\top X$ is never assembled. Each CG iteration requires only two matrix-vector products -with $X$, reducing per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. Long-only weights are +%That the minimum-variance solution satisfies $w \propto \Sigma^{-1}\mathbf{1}$, where $\mathbf{1}$ denotes a conformable vector of ones, +%is a classical observation: it follows +%directly from the KKT stationarity condition $2\Sigma w = \lambda\mathbf{1}$ +%\tcr{At this point, many readers will not know what is meant by KKT.} +%and was made explicit by +%\cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the single rescaling +%$w = v/(\mathbf{1}^\top v)$ where $v = \Sigma^{-1}\mathbf{1}$. What has \emph{not} been done is to exploit this +%structure computationally: the standard pipeline still forms $\Sigma = X^\top X/T$ explicitly before solving. + +This paper makes two contributions. First, we apply conjugate gradients (CG) to $\hat \Sigma v = \mathbf{1}$ +\emph{matrix free}: $\hat \Sigma = X^\top X$ never needs to be assembled explicitly. Each CG iteration requires only two matrix-vector products +with $X$, reducing per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. Non-negative weights are enforced by a primal-dual active-set loop that calls the CG solver on successively smaller subproblems. -Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004} plays a dual role in this framework. Statistically, it +Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004}, a method suggest to come up with a more accurate estimator of the true covariance matrix $\Sigma$ compared to the sample covariance matrix, +plays a dual role in this framework. Statistically, it reduces estimation error by shrinking sample eigenvalues toward their mean. Numerically, the same modification -compresses the eigenvalue spread of the system matrix and directly accelerates CG convergence: on S\&P~500 equity data, -shrinkage cuts CG iterations from 744 to 82. Statistical regularisation and spectral preconditioning are, in this +compresses the eigenvalue spread of the system matrix and directly accelerates CG convergence. For example, on S\&P~500 equity data, using LW shrinkage instead of the sample covariance matrix reduces the number of CG iterations from 744 to 82. +Statistical regularization and spectral preconditioning are, in this problem, the same operation applied to the same matrix---a connection that does not appear to have been made explicit in the existing literature. -CG~\cite{hestenes1952} is a cornerstone of large-scale scientific computing~\cite{golub2013} but has not been applied +CG of Hestenes and Stiefel \cite{hestenes1952} is a cornerstone of large-scale scientific computing; e.g., see Golub and van Loan \cite{golub2013}. However, to the best of our knowledge it has not been +applied yet to portfolio optimisation in the matrix-free form described here. The closest related work applies GMRES to the minimum -variance problem when $\Sigma$ is singular~\cite{bajeux2012}; that formulation still assembles $X^\top X$ and does not +variance problem when $\hat \Sigma$ is singular~\cite{bajeux2012}; that formulation still assembles $X^\top X$ and does not exploit the SPD structure or consider preconditioning. General-purpose solvers such as -Clarabel~\cite{clarabel2024} handle the long-only constraint natively but require an explicit covariance matrix. -ywa -Sections~\ref{sec:problem}--\ref{sec:precond} develop the formulation and solvers; Section~\ref{sec:results} reports +Clarabel~\cite{clarabel2024} handle the long-only constraint natively +but require as input an estimate of the covariance matrix. +The remainder of the paper is organized as follows. +\mbox{Sections~\ref{sec:problem}--\ref{sec:precond}} develop the formulation and solvers; Section~\ref{sec:results} reports empirical results on both synthetic and S\&P~500 data. \section{Problem Formulation}\label{sec:problem} - -Let $X \in \mathbb{R}^{T \times n}$ be the demeaned return matrix for $n$ assets over $T$ trading days. The long-only -minimum variance problem is: +The long-only mean-variance portfolio solves +\[ +\min_{w}\; w^\top\Sigma\, w - \rho\,\mu^\top w +\quad\text{subject to}\quad +\mathbf{1}^\top w = 1,\quad w\geq 0~, +\] +where $\Sigma$ is the $n\times n$ covariance matrix of returns, $\mu$ is the $n\times 1$ vector of expected returns, and $\rho\geq 0$ controls the risk-return trade-off; setting $\rho=0$ recovers the pure long-only minimum variance portfolio. +Since both $\Sigma$ and +$\mu$ are unknown, in practice they must be replaced by estimates. Let $X\in\mathbb{R}^{T\times n}$ denote the demeaned return matrix for $n$ assets over the previous $T$ trading periods. The feasible problem formulation then is \[ -\min_{w} \; w^\top \hat\Sigma\, w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, \quad w \geq 0 +\min_{w}\; w^\top\hat\Sigma\, w - \rho\,\hat\mu^\top w +\quad\text{subject to}\quad +\mathbf{1}^\top w = 1,\quad w\geq 0~. \] -where $\hat\Sigma = \frac{1}{T} X^\top X$ is the sample covariance matrix. This is equivalently stated as: +The classical choice for estimating $\Sigma$ +the sample covariance matrix $\hat\Sigma = X^\top X/T$, for which $w^\top\hat\Sigma\, w = \norm{Xw}^2/T$, and the problem becomes \[ -\min_{w} \; \frac{\|Xw\|^2}{T} = \min_{w}\; w^\top \hat\Sigma\, w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, \quad w \geq 0 +\min_{w}\;\frac{\norm{Xw}^2}{T} - \rho\,\hat\mu^\top w +\quad\text{subject to}\quad +\mathbf{1}^\top w = 1,\quad w\geq 0~. \] -This reframing is key: the objective is now a normalised squared norm of $Xw$, and we can evaluate -$w \mapsto X^\top(Xw)$ using two matrix-vector products, without ever forming $X^\top X$. +This reframing is key: the variance term is now a normalised squared norm of $Xw$, and the gradient $w\mapsto X^\top(Xw)$ can be evaluated with two matrix--vector products, without ever forming $X^\top X$. \begin{table}[ht] \centering @@ -121,8 +169,6 @@ \section{Problem Formulation}\label{sec:problem} Parameter & Type & Description \\ \midrule $X$ & $\mathbb{R}^{T \times n}$ & Demeaned return matrix \\ -$\texttt{target}$ & $\mathbb{R}^{n \times n}$ & Shrinkage target matrix (optional; default \texttt{None}) \\ -$\alpha$ & $[0,1]$ & Shrinkage intensity; active only when \texttt{target} is provided (default $0$) \\ $\rho$ & $\mathbb{R}_{\geq 0}$ & Return-tilt weight (default $0$) \\ $\mu$ & $\mathbb{R}^n$ & Expected returns (default $\mathbf{0}$) \\ \bottomrule @@ -131,163 +177,189 @@ \section{Problem Formulation}\label{sec:problem} \label{tab:params} \end{table} -\paragraph{Dual problem.} +\paragraph{Dual problem} Including multipliers for both constraints, the full Lagrangian is \[ - \mathcal{L}(w, \lambda, \nu) - = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1) - \nu^\top w, +\mathcal{L}(w,\, \lambda,\, \nu) += \frac{\lVert Xw\rVert^2}{T} - \rho\hat\mu^\top w + - \lambda(\mathbf{1}^\top w - 1) - \nu^\top w\;, \] -where $\lambda\in\mathbb{R}$ is the budget multiplier and $\nu\geq\mathbf{0}$ is the vector of non-negativity -multipliers. Minimising over $w$ via stationarity $\frac{2}{T} X^\top X w = \lambda\mathbf{1}+\nu$ gives the dual function +where $\lambda\in\mathbb{R}$ is the budget multiplier and $\nu\geq\mathbf{0}$ is the vector of +non-negativity multipliers. The stationarity condition \[ - g(\lambda,\nu) - = \inf_w \mathcal{L}(w,\lambda,\nu) - = \lambda - \tfrac{T}{4}(\lambda\mathbf{1}+\nu)^\top(X^\top X)^{-1}(\lambda\mathbf{1}+\nu), +\tfrac{2}{T}\, X^\top X w = \lambda\mathbf{1}+\nu+\rho\hat\mu \] -defined when $X$ has full column rank; the dual problem is $\max_{\lambda,\,\nu\geq\mathbf{0}}\,g(\lambda,\nu)$. The -feasible set $\{w:\mathbf{1}^\top w=1,\,w\geq\mathbf{0}\}$ is compact and $w=\tfrac{1}{n}\mathbf{1}$ is strictly -feasible, so Slater's condition holds and strong duality obtains: the primal and dual optima coincide and the duality -gap is zero. - -At the optimum, complementary slackness ($\nu_i w_i = 0$ for all $i$) partitions assets into two types. \emph{Active} -assets ($w_i>0$) have $\nu_i=0$; stationarity then gives $\frac{2}{T}(X^\top Xw)_i=\lambda$, so every active asset -contributes equally to marginal portfolio variance. \emph{Excluded} assets ($w_i=0$) have -$\nu_i = \frac{2}{T}(X^\top Xw)_i - \lambda \geq 0$: -their marginal variance is at least $\lambda$ and they would increase portfolio variance if held at a positive level. -The primal--dual loop in Section~\ref{sec:kkt} is a direct implementation of these conditions: the loop terminates -precisely when both primal feasibility and dual feasibility hold, which---together with stationarity from each inner -solve---is sufficient for global optimality. - -\subsection*{Problem variants.} - -The framework handles a range of standard portfolio construction problems directly. Each is expressed as -\begin{equation} - \min_{w}\;\|\tilde{X}w\|^2 - \rho\mu^\top w - \quad\text{subject to}\quad - \mathbf{1}^\top w = 1,\;\; w \geq 0, - \label{eq:general} -\end{equation} -through choices of $\tilde{X}$, $\rho$, and $\mu$ summarised in Table~\ref{tab:problems}. No change to the -algorithm is required; only the data passed to the solver changes. - -\begin{table}[ht] -\centering -\begin{tabular}{llll} -\toprule -Portfolio problem & $\tilde{X}$ & $\rho$ & $\mu$ \\ -\midrule -Minimum variance & $X$ & $0$ & --- \\ -Mean-variance (Markowitz) & $X$ & $>0$ & expected returns \\ -Minimum tracking error$^*$ & $X$ & $2$ & $X^\top(Xb)$ \\ -LW-regularised minimum variance & $[\sqrt{(1-\alpha)/T}\,X;\,\sqrt{\alpha}\,R]$ & $0$ & --- \\ -Factor-model covariance $FF^\top+D$ & $[F;\,D^{1/2}]$ & $0$ & --- \\ -Regime-blended / multi-period & $[\omega_1 X_1;\,\ldots;\,\omega_K X_K]$ & $0$ & --- \\ -\bottomrule -\end{tabular} -\caption{Portfolio construction problems solved by the same primal--dual loop. - $b$ is the benchmark weight vector; $F$ contains factor returns; $D$ is a diagonal specific-risk matrix; - $R$ is the $N\times N$ regularisation target matrix (e.g.\ $\sqrt{\bar\lambda}\,I$ for classical Ledoit-Wolf - shrinkage, where $\bar\lambda = \|X\|_F^2/(nT)$ is the mean eigenvalue of $\hat\Sigma$). - Stacking of $\tilde{X}$ is in the row direction. - $^*\min_w (w-b)^\top\Sigma(w-b) = \min_w w^\top\Sigma w - 2(\Sigma b)^\top w + \mathrm{const}$, so - $\rho\mu = 2\Sigma b = 2X^\top(Xb)$ (two matrix--vector products). - Since $\mu = \Sigma b$, the second SPD solve yields $v_2 = b$ exactly; tracking error adds no inner-solve cost.} -\label{tab:problems} -\end{table} - -\paragraph{What does not fit.} -The maximum Sharpe ratio (tangency) portfolio $\arg\max_w \mu^\top w / \sqrt{w^\top\Sigma w}$, subject to -$w \geq 0$, is a fractional programme and does not reduce to \eqref{eq:general} with a fixed $\rho$. It can be -recovered parametrically by sweeping $\rho$ over the efficient frontier and locating the tangency point, but this -requires multiple solves. Problems with additional linear inequality constraints---sector limits, turnover -bounds, factor-exposure limits---also lie outside the current template; a general QP solver (e.g.\ CVXPY with -Clarabel) is the appropriate tool for those. - -The one structural requirement is that the covariance admit a factorisation $\Sigma = \tilde{X}^\top\tilde{X}$ -with $\tilde{X}$ available directly or by cheap row-augmentation, so that the matrix-vector product -$v\mapsto\tilde{X}^\top(\tilde{X}v)$ remains $O(T'n)$ per iteration. Nonlinear shrinkage~\cite{ledoit2020}, -which applies a separate coefficient to each sample eigenvalue, does not admit such a factorisation and requires -a different preconditioning strategy. - -\section{Positive Definite Reduction and Primal-Dual Loop}\label{sec:kkt} +gives $w^* = \tfrac{T}{2}(X^\top X)^{-1}s$ with $s = \lambda\mathbf{1}+\nu+\rho\hat\mu$. +Substituting into $\mathcal{L}$: +\[ +\mathcal{L}(w^*,\,\lambda,\,\nu) += \tfrac{1}{T}(w^*)^\top X^\top X w^* - s^\top w^* + \lambda\;. +\] +From stationarity $\tfrac{1}{T}X^\top X w^* = \tfrac{s}{2}$, so the first term equals +$\tfrac{1}{2}s^\top w^*$, giving +\[ +g(\lambda,\,\nu) += \tfrac{1}{2}s^\top w^* - s^\top w^* + \lambda += -\tfrac{1}{2}s^\top w^* + \lambda\;. +\] +Substituting $w^* = \tfrac{T}{2}(X^\top X)^{-1}s$ yields the dual function +\[ +g(\lambda,\,\nu) += \inf_w\; \mathcal{L}(w,\,\lambda,\,\nu) += \lambda - \tfrac{T}{4}\, + (\lambda\mathbf{1}+\nu+\rho\hat\mu)^\top(X^\top X)^{-1} + (\lambda\mathbf{1}+\nu+\rho\hat\mu)\;, +\] +defined when $X$ has full column rank; the dual problem is +$\max_{\lambda,\,\nu\geq\mathbf{0}}\; g(\lambda,\,\nu)$. +The feasible set $\{w:\mathbf{1}^\top w=1,\,w\geq\mathbf{0}\}$ is compact and +$w=\tfrac{1}{n}\mathbf{1}$ is strictly feasible, so Slater's condition holds and +strong duality obtains. + +At the optimum, complementary slackness ($\nu_i w_i = 0$ for all $i$) partitions assets +into two types. \emph{Active} assets ($w_i>0$) have $\nu_i=0$; stationarity then gives +\[ +\tfrac{2}{T}\,(X^\top Xw)_i = \lambda+\rho\hat\mu_i\;, +\] +equating each active asset's marginal variance to the budget shadow price adjusted for its +expected return. \emph{Excluded} assets ($w_i=0$) satisfy +\[ +\nu_i = \tfrac{2}{T}\,(X^\top Xw)_i - \lambda - \rho\hat\mu_i \geq 0~. +\] +Their marginal variance exceeds this threshold, so holding them at a positive weight would +increase variance net of return. The primal--dual loop in Section~\ref{sec:primaldual} is a direct +implementation of these conditions: it terminates precisely when primal and dual feasibility +both hold, which---together with stationarity from each inner solve---is sufficient for +global optimality. + +%\subsection*{Problem variants} +% +%The framework handles a range of standard portfolio construction %problems directly. Each is expressed as +%\begin{equation} +% \min_{w}\;\|\tilde{X}w\|^2 - \rho\mu^\top w +% \quad\text{subject to}\quad +% \mathbf{1}^\top w = 1,\;\; w \geq 0~, +% \label{eq:general} +%\end{equation} +%through choices of $\tilde{X}$, $\rho$, and $\mu$ summarized in %Table~\ref{tab:problems}. No change in the +%algorithm is required; only the data passed to the solver change. + +%\begin{table}[ht] +%\centering +%\begin{tabular}{llll} +%\toprule +%Portfolio problem & $\tilde{X}$ & $\rho$ & $\mu$ \\ +%\midrule +%Minimum variance & $X$ & $0$ & --- \\ +%Mean-variance (Markowitz) & $X$ & $>0$ & expected returns \\ +%Minimum tracking error$^*$ & $X$ & $2$ & $X^\top(Xb)$ \\ +%LW-regularized minimum variance & $[\sqrt{(1-%\alpha)/T}\,X;\,\sqrt{\alpha}\,R]$ & $0$ & --- \\ +%Factor-model covariance $FF^\top+D$ & $[F;\,D^{1/2}]$ & $0$ & --- \\ +%Regime-blended / multi-period & $[\omega_1 X_1;\,\ldots;\,\omega_K %X_K]$ & $0$ & --- \\ +%\bottomrule +%\end{tabular} +%\caption{Portfolio construction problems solved by the same primal--dual loop. +% $b$ is the benchmark weight vector; $F$ contains factor returns; $D$ is a diagonal specific-risk matrix; +% $R$ is the $N\times N$ regularisation target matrix (e.g.\ $\sqrt{\bar\lambda}\,I$ for classical Ledoit-Wolf +% shrinkage, where $\bar\lambda = \|X\|_F^2/(nT)$ is the mean eigenvalue of $\hat\Sigma$). +% Stacking of $\tilde{X}$ is in the row direction. +% $^*\min_w (w-b)^\top\Sigma(w-b) = \min_w w^\top\Sigma w - 2(\Sigma b)^\top w + \mathrm{const}$, so +% $\rho\mu = 2\Sigma b = 2X^\top(Xb)$ (two matrix--vector products). +% Since $\mu = \Sigma b$, the second SPD solve yields $v_2 = b$ exactly; tracking error adds no inner-solve cost.} +%\label{tab:problems} +%\end{table} + +%\paragraph{What does not fit} +%The maximum Sharpe ratio portfolio, also called the tangency portfolio, solves +%\[ +%\arg\max_w\; \frac{\mu^\top w}{\sqrt{w^\top\Sigma w}} +%\quad\text{subject to}\quad \mathbf{1}^\top w = 1,\; w\geq 0. +%\] +%This is a fractional programme and does not reduce to \eqref{eq:general} with a fixed $\rho$. +%It can be +%recovered parametrically by sweeping $\rho$ over the efficient frontier and locating the tangency point, but this +%requires multiple solves. Problems with additional linear inequality constraints---sector limits, turnover +%bounds, factor-exposure limits---also lie outside the current template; a general QP solver (e.g.\ CVXPY with +%Clarabel) is the appropriate tool for those. + +%The one structural requirement is that the covariance admit a factorization $\Sigma = \tilde{X}^\top\tilde{X}$ +%with $\tilde{X}$ available directly or by cheap row-augmentation, so that the matrix-vector product +%$v\mapsto\tilde{X}^\top(\tilde{X}v)$ remains $O(T'n)$ per iteration. Nonlinear shrinkage~\cite{ledoit2020}, +%which applies a separate coefficient to each sample eigenvalue, does not admit such a factorization and requires +%a different preconditioning strategy. + +\section{Positive-Definite Reduction}\label{sec:pd} Ignoring the inequality constraints for now, the Lagrangian for the equality-constrained problem is: \[ -\mathcal{L}(w, \lambda) = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1) +\mathcal{L}(w, \lambda) = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1)~ + \rho \hat \mu~. \] Setting the derivative to zero gives the stationarity condition -$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\mu$, where $\hat\Sigma = X^\top X/T$ (or $\Sigma_{\mathrm{LW}}$ with shrinkage). -For $\rho = 0$ this reduces to $\frac{2}{T}\hat\Sigma\, w = \lambda\mathbf{1}$, determining the solution up to a scalar. +$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\hat \mu$, where $\hat\Sigma = X^\top X/T$ (or $\hat \Sigma_{\mathrm{LW}}$ with shrinkage). +For $\rho = 0$, this reduces to $\frac{2}{T}\hat\Sigma\, w = \lambda\mathbf{1}$, determining the solution up to a scalar. -\paragraph{Positive definite reduction.} -When $\rho = 0$ (pure minimum variance), stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1}$, +\paragraph{Positive-definite reduction} +When $\rho = 0$ (pure long-only minimum variance), stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1}$, so $w$ is a scalar multiple of $\hat\Sigma^{-1}\mathbf{1}$. Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and writing $w = c\,v_1$, the budget constraint -$\mathbf{1}^\top w = 1$ fixes +$\mathbf{1}^\top w = 1$ \tcb{scales} \[ - c = \frac{1}{\mathbf{1}^\top v_1}, \qquad w = \frac{v_1}{\mathbf{1}^\top v_1}. + c = \frac{1}{\mathbf{1}^\top v_1} \quad \mbox{and} \quad w = \frac{v_1}{\mathbf{1}^\top v_1}~. \] -When $\rho \neq 0$, stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1} + \rho\mu$, so -$w = \tfrac{\lambda}{2}\,\hat\Sigma^{-1}\mathbf{1} + \tfrac{\rho}{2}\,\hat\Sigma^{-1}\mu$. -Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and $v_2 = \hat\Sigma^{-1}\mu$ (two SPD solves), the budget +When $\rho \neq 0$, stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1} + \rho\hat \mu$, so +$w = \tfrac{\lambda}{2}\,\hat\Sigma^{-1}\mathbf{1} + \tfrac{\rho}{2}\,\hat\Sigma^{-1}\hat \mu$. +Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and $v_2 = \hat\Sigma^{-1}\hat \mu$ (two SPD solves), the budget constraint determines the multiplier \[ - \lambda = \frac{2 - \rho\,(\mathbf{1}^\top v_2)}{\mathbf{1}^\top v_1}, + \lambda = \frac{2 - \rho\,(\mathbf{1}^\top v_2)}{\mathbf{1}^\top v_1}~, \] and the weights follow as $w = \tfrac{\lambda}{2}\,v_1 + \tfrac{\rho}{2}\,v_2$. In both cases the $(n+1)\times(n+1)$ saddle-point system is never needed and $\lambda$ is recovered analytically rather than as part of the solve. -When $\alpha > 0$ and a regularisation target $R$ is provided, the regularised matrix -$\Sigma_{\mathrm{LW}} = \frac{1-\alpha}{T}\,X^\top X + \alpha\,R^\top R = (1-\alpha)\,\hat\Sigma + \alpha\,R^\top R$ -is symmetric positive definite and standard CG applies directly to $\Sigma_{\mathrm{LW}}\,v = \mathbf{1}$. -For classical Ledoit-Wolf shrinkage one sets $R = \sqrt{\bar\lambda}\,I$ with $\bar\lambda = \|X\|_F^2/(nT)$, -giving $\Sigma_{\mathrm{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\bar\lambda\,I$. -For completeness, the equivalent $(n+1)\times(n+1)$ saddle-point formulation -\[ -\begin{pmatrix} 2\Sigma_{\mathrm{LW}} & \mathbf{1} \\ \mathbf{1}^\top & 0 \end{pmatrix} -\begin{pmatrix} w \\ \lambda \end{pmatrix} -= -\begin{pmatrix} \mathbf{0} \\ 1 \end{pmatrix} -\] -is symmetric but indefinite and requires MINRES \cite{paige1975} rather than CG; it is shown here for completeness. +\section{Primal-Dual Loop}\label{sec:primaldual} -The long-only constraint $w \geq 0$ is enforced by a \textbf{primal-dual iterative procedure}. Each outer step solves -the equality-constrained (budget-only) $n_a \times n_a$ positive definite system over the current asset universe. In the -\emph{primal step}, any asset whose weight is negative is dropped. Once all active weights are non-negative, a -\emph{dual step} checks the KKT gradient condition for every excluded asset: for global optimality, every excluded asset -$i$ must satisfy +The long-only constraint $w \geq 0$ is enforced by a \emph{primal--dual iterative procedure}. +Each outer step solves the equality-constrained (budget-only) $n_a \times n_a$ positive +definite system over the current active set. In the \emph{primal step}, any asset whose weight +is negative is dropped. Once all active weights are non-negative, a \emph{dual step} checks +the KKT gradient condition for every excluded asset: for global optimality, every excluded +asset $i$ must satisfy \[ - \nabla_i f(w) \;\geq\; \lambda, - \qquad - \nabla_i f(w) = 2\Bigl(\tfrac{1-\alpha}{T}(X^\top X w)_i + \alpha\,(R^\top R\,w)_i\Bigr) - \rho\mu_i, +\nabla_i f(w) \;\geq\; \lambda\;, +\quad\text{and}\quad +\nabla_i f(w) = \tfrac{2}{T}(X^\top X w)_i - \rho\hat\mu_i\;, \] -where $\lambda$ is the budget Lagrange multiplier and $R$ is the regularisation target matrix. An excluded asset that fails this -condition would decrease portfolio variance if added at a small positive level; it is re-added to the active set and the -loop repeats. The loop terminates when both primal and dual feasibility hold simultaneously; combined with stationarity -from the KKT solve, this is sufficient for global optimality. The feasible set is non-empty and compact, so the -constrained minimum exists. +where $\lambda$ is the budget Lagrange multiplier. An excluded asset that fails this condition +would improve the objective if added at a small positive level; it is re-added to the active +set and the loop repeats. The loop terminates when both primal and dual feasibility hold +simultaneously; combined with stationarity from the inner solve, this is sufficient for +global optimality. + + +\tcb{This comes all a bit too early: With Ledoit-Wolf shrinkage, the asset elimination loop converges rapidly in practice. Let $\bar\lambda = \|X\|_F^2/(nT)$ -denote the average eigenvalue of the sample covariance. The unconstrained minimiser satisfies $w^\star \propto -\Sigma_{\mathrm{LW}}^{-1}\mathbf{1}$. As $\alpha \to 1$, $\Sigma_{\mathrm{LW}} \to \bar\lambda I$ and $w^\star \to -\frac{1}{n}\mathbf{1}$, the equal-weight portfolio with all components strictly positive. By continuity there exists a +denote the average eigenvalue of the sample covariance. The unconstrained minimizer satisfies $w^\star \propto +\hat \Sigma_{\mathrm{LW}}^{-1}\mathbf{1}$. As $\alpha \to 1$, $\hat \Sigma_{\mathrm{LW}} \to \bar\lambda I$ and $w^\star \to +\frac{1}{n}\mathbf{1}$, the equal-weighted $1/n$ portfolio with all components strictly positive. By continuity, there exists a threshold $\alpha^\star \in (0,1)$ above which every component of the unconstrained solution is positive; for $\alpha > \alpha^\star$ the first solve requires no asset to be dropped. The first-order expansion \[ -\Sigma_{\mathrm{LW}}^{-1}\mathbf{1} \approx \tfrac{1}{\alpha\bar\lambda}\mathbf{1} - -\tfrac{1-\alpha}{\alpha^2\bar\lambda^2 T}X^\top X\mathbf{1}, +\hat \Sigma_{\mathrm{LW}}^{-1}\mathbf{1} \approx \tfrac{1}{\alpha\bar\lambda}\mathbf{1} - +\tfrac{1-\alpha}{\alpha^2\bar\lambda^2 T}X^\top X\mathbf{1}~, \] valid for $\alpha$ close to one, shows that $\alpha^\star$ is governed by the spectral structure of $X^\top X$: assets -with large total covariance exposure are the last to turn positive as $\alpha$ increases. For the LW-optimal $\alpha$ -and dense equity returns, the threshold is typically exceeded and the first solve yields few negative components. Across +with large total covariance exposure are the last to turn positive as $\alpha$ increases. For the LW-optimal $\alpha$ \tcr{A brief explanation would be needed here, optimal in what sense?} +and dense equity returns, the threshold is typically exceeded \tcr{This is an empirical finding, I guess? There cannot be an theory for such a finding} and the first solve yields few negative components. Across 50 experiments with $n$ from 100 to 1000 and $T = 2n$, the elimination loop required 2--3 outer iterations in all cases; -runtime is dominated by the linear solves. +runtime is dominated by the linear solves. \tcr{Some reference to such experiments would be useful, that statement comes a bit ``out of the blue", without underlying details.} +} \begin{algorithm}[ht] \caption{Primal-dual outer loop}\label{alg:elim} \begin{algorithmic}[1] -\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$;\; regularisation target $R$ (optional) +\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$; \Ensure Weight vector $w \in \mathbb{R}^n$;\; total inner iteration count $k$ \State $\mathrm{active} \leftarrow \{1,\ldots,n\}$;\; $k \leftarrow 0$ \Loop @@ -297,7 +369,7 @@ \section{Positive Definite Reduction and Primal-Dual Loop}\label{sec:kkt} \hfill\textit{(primal step: drop violators)} \Else \State $w_i \leftarrow w_{a,i}$ for $i \in \mathrm{active}$;\; $w_i \leftarrow 0$ for $i \notin \mathrm{active}$ - \State $g_i \leftarrow 2\bigl(\tfrac{1-\alpha}{T}(X^\top X w)_i + \alpha\,(R^\top R\,w)_i\bigr) - \rho\mu_i$ + \State $g_i \leftarrow \tfrac{2}{T}(X^\top X w)_i - \rho\hat \mu_i$ \State $\lambda \leftarrow \mathrm{mean}_{j \in \mathrm{active}}(g_j)$ \State $\mathcal{V} \leftarrow \{i \notin \mathrm{active} : g_i < \lambda - \varepsilon\}$ \If{$\mathcal{V} = \emptyset$} @@ -311,80 +383,113 @@ \section{Positive Definite Reduction and Primal-Dual Loop}\label{sec:kkt} \end{algorithmic} \end{algorithm} -Algorithm~\ref{alg:elim} accepts any inner solver $f$; Section~\ref{sec:solvers} presents four concrete choices: -matrix-free CG (the primary method), a direct dense factorisation, non-negative least squares (NNLS), and the CVXPY interior-point reference. +Algorithm~\ref{alg:elim} accepts any inner solver $f$; Section~\ref{sec:solvers} presents concrete choices for $f$. + \section{Computational Methods}\label{sec:solvers} \subsection{CG Method}\label{ssec:cg} -The inner solver $f$ for Algorithm~\ref{alg:elim} is conjugate gradients applied to the $n_a \times n_a$ SPD system -derived in Section~\ref{sec:kkt}. The action of $\Sigma_{\mathrm{LW}}$ on a vector requires only two matrix-vector -products with $X_{\mathrm{a}}$; the matrix itself is never formed. +The inner solver $f$ for Algorithm~\ref{alg:elim} is conjugate gradients applied to the +$n_a \times n_a$ SPD system derived in Section~\ref{sec:pd}. The action of $\hat\Sigma_a$ +on a vector requires only two matrix-vector products with $\tilde{X}_a$; the matrix itself +is never formed. \begin{algorithm}[ht] \caption{CG inner step (inner solver $f$ for Algorithm~\ref{alg:elim})}\label{alg:cg} \begin{algorithmic}[1] -\Require Active-asset index set $\mathrm{active}$;\; problem data;\; regularisation target $R$ (optional) +\Require Active-asset index set $\mathrm{active}$;\; problem data $\tilde{X}$ \Ensure Active-asset weights $w_{\mathrm{a}}$;\; CG iteration count $k$ \Function{CgSolve}{active} -\State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $R_{\mathrm{a}} \leftarrow R[\mathrm{active},\mathrm{active}]$,\; $n_a \leftarrow - |\mathrm{active}|$ - \State Build matrix-free operator $\mathcal{S}$ of size $n_a \times n_a$: - \Statex\hspace{2em} $v \;\mapsto\; \tfrac{1-\alpha}{T}\,X_{\mathrm{a}}^\top(X_{\mathrm{a}}\,v) + \alpha\,R_{\mathrm{a}}^\top(R_{\mathrm{a}}\,v)$ - \hfill\textit{($\Sigma_{\mathrm{LW}}$ applied matrix-free)} - \State Solve $\mathcal{S}\,v = \mathbf{1}$ via CG - \State \Return $v\,/\,(\mathbf{1}^\top v)$,\; iteration count +\State Let $\tilde{X}_{\mathrm{a}} \leftarrow \tilde{X}_{:,\mathrm{active}}$,\; + $n_a \leftarrow |\mathrm{active}|$ +\State Build matrix-free operator $\mathcal{S}$ of size $n_a \times n_a$: + \Statex\hspace{2em} + $v \;\mapsto\; \tilde{X}_{\mathrm{a}}^\top(\tilde{X}_{\mathrm{a}}\,v)$ + \hfill\textit{($\hat\Sigma_a$ applied matrix-free)} +\State Solve $\mathcal{S}\,v = \mathbf{1}$ via CG +\State \Return $v\,/\,(\mathbf{1}^\top v)$,\; iteration count \EndFunction \end{algorithmic} \end{algorithm} -When an asset is dropped in the primal step, $\mathcal{S}$ is rebuilt for the reduced active set and CG restarts from -scratch on the smaller system. +When an asset is dropped in the primal step, $\mathcal{S}$ is rebuilt for the reduced active +set and CG restarts from scratch on the smaller system. -\paragraph{Convergence speed.} +\paragraph{Convergence speed} The classical CG error bound \cite{greenbaum1997} states \[ - \frac{\|e_k\|_{\Sigma_{\mathrm{LW}}}}{\|e_0\|_{\Sigma_{\mathrm{LW}}}} - \;\leq\; - 2\!\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{\!k}, - \qquad - \kappa = \kappa(\Sigma_{\mathrm{LW}}), +\frac{\lVert e_k\rVert_{\hat\Sigma_a}}{\lVert e_0\rVert_{\hat\Sigma_a}} +\;\leq\; +2\!\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{\!k}\;, +\qquad \kappa = \kappa(\hat\Sigma_a)\;, \] -where $\|e\|_{\Sigma} = (e^\top \Sigma e)^{1/2}$ is the energy norm and $\kappa$ is the spectral condition number -introduced in Section~\ref{sec:precond}. Shrinkage compresses $\kappa$ monotonically as $\alpha$ increases, directly -improving the asymptotic convergence rate. +where $\lVert e\rVert_{\hat\Sigma_a} = (e^\top\hat\Sigma_a\, e)^{1/2}$ denotes the energy norm +and $\kappa$ is the spectral condition number of the active-asset submatrix. + -A second, sharper bound follows from the rank structure of $\Sigma_{\mathrm{LW}}$. +\tcb{That's just no required and wrong: +A second, sharper bound follows from the rank structure of $\hat \Sigma_{\mathrm{LW}}$. Because $X_{\mathrm{a}} \in \mathbb{R}^{T \times n_a}$ has rank at most $\min(T,n_a)$, the matrix -$\Sigma_{\mathrm{LW}} = \frac{1-\alpha}{T}X_{\mathrm{a}}^\top X_{\mathrm{a}} + \alpha R_{\mathrm{a}}^\top R_{\mathrm{a}}$ +$\hat \Sigma_{\mathrm{LW}} = \frac{1-\alpha}{T}X_{\mathrm{a}}^\top X_{\mathrm{a}} + \alpha R_{\mathrm{a}}^\top R_{\mathrm{a}}$ has \emph{at most} $\min(T,n_a)+1$ distinct eigenvalues: exactly $n_a - \min(T,n_a)$ eigenvalues equal to -$\alpha\lambda_{\min}(R_{\mathrm{a}}^\top R_{\mathrm{a}})$ (the ridge floor) and up to $\min(T,n_a)$ distinct eigenvalues above it. +$\alpha\lambda_{\min}(R_{\mathrm{a}}^\top R_{\mathrm{a}})$ (the ridge floor) and up to $\min(T,n_a)$ distinct eigenvalues above it. \tcr{I do not quite understand. $\hat \Sigma_{\mathrm{LW}}$ has as many distinct eigenvalues as the sample covariance matrix, which have $n$ distinct eigenvalues with probability equal to one if the vector of asset returns has continuous distribution, and as long as $T > n$.} Since CG terminates in exact arithmetic in at most as many steps as there are distinct eigenvalues -\cite{greenbaum1997}, convergence requires at most $\min(T,n_a)+1$ iterations. -In the underdetermined regime $T < n_a$ this yields a bound of $T+1$, independent of the active-set size~$n_a$; +(Greenbaum \cite{greenbaum1997}), convergence requires at most $\min(T,n_a)+1$ iterations. +In the underdetermined regime $T < n_a$, this yields a bound of $T+1$, independent of the active-set size~$n_a$; adding assets to the portfolio does not increase the iteration count. This regime arises when the history window is shorter than the universe (a common scenario in, e.g., high-frequency or short-window portfolio rebalancing), or within the primal-dual loop once enough assets have been -excluded that $n_a$ exceeds $T$. Shrinkage reinforces the rank-based bound by clustering the $\min(T,n_a)$ -non-ridge eigenvalues, causing super-linear acceleration in practice \cite{simoncini2013}. +excluded that $n_a$ exceeds~$T$. Shrinkage reinforces the rank-based bound by clustering the $\min(T,n_a)$ +non-ridge eigenvalues, causing super-linear acceleration in practice; e.g., see +Simoncini and Szyld \cite{simoncini2013}. The iteration counts in Table~\ref{tab:sp500} illustrate the condition-number effect: -on the S\&P~500 problem ($n=495$, $T=1192 > n$) shrinkage reduces CG iterations from 744 to 82, consistent -with $\kappa(\Sigma_{\mathrm{LW}})$ decreasing monotonically as $\alpha$ increases (Section~\ref{sec:precond}). +On the S\&P~500 problem ($n=495$, $T=1192 > n$), using LW shrinkage instead of the sample covariance matrix +reduces CG iterations from 744 to 82, which is consistent +with $\kappa(\hat \Sigma_{\mathrm{LW}})$ decreasing monotonically as $\alpha$ increases (Section~\ref{sec:precond}).} +From the discussion above, it becomes clear that the distribution of the eigenvalues of the linear system is crucial for achieving fast convergence of the conjugate gradient method (cf. \cite{saad2003iterative}). One way of improving the convergence speed to solve the linear system $Ax=b$ is to use a preconditioner $M$ and the modified system $M^{-1}Ax=M^{-1}b,$ where the new system matrix $M^{-1}A$ has a more favorable distribution of the eigenvalues. The two main objectives for designing the preconditioner $M$ are that it somehow closely resembles $A$ but is also easy to apply. Preconditioners based on sketching \cite{meng2014lsrn,frangella2023randomized} are used for shifted symmetric positive definite systems. In spirit similar to what we suggest next is the method introduced in \cite{avron2010blendenpik} where a preconditioned matrix $XR^{-1}$ is fed into the LSQR solver \cite{paige1982algorithm}, where $R$ comes from a QR factorization. + +% \tcb{Should we have the cg algorithm here? A few words about preconditioning? Martin, don't hold back} + +\subsection*{Sherman-Morrison-Woodbury Preconditioner} +One possibility for preconditioning the system matrix $A = \omega X_a^\top X_a + \gamma I$ is to use a low-rank approximation of the matrix $X_a$ and then apply the Sherman-Morrison-Woodbury \cite{golub2013} identity +$ +\left(I+UCV\right)^{-1}=I-U\left(C^{-1}+VU\right)^{-1}V. +$ +We then utilize the idea that $UCV$ is a low-rank approximation to a large square matrix and that $\left(C^{-1}+VU\right)$ is a small matrix. If additionally $V=U^\top$ is a matrix with orthogonal columns we get +$\left(I+UCV\right)^{-1}=I-U\left(C^{-1}+I\right)^{-1}U^\top.$ +We now perform the low-rank approximation of $X_a$ using the truncated singular value decomposition. Obviously, the quality of such an approximation depends on the singular value decay of $X_a$. We start by factoring out $\gamma$ and substitute the rank-$k$ SVD of $X_a \approx U_k S_k V_k^\top$ +to get the approximation +\begin{equation} + A \approx \underbrace{\gamma\!\left(I + V_k \underbrace{\tfrac{\omega}{\gamma}S_k^2}_{C} V_k^\top\right)}_{M}~. +\end{equation} +Since $V_k^\top V_k = I_k$, we now obtain the preconditioner +\begin{equation} + M^{-1} = \frac{1}{\gamma}\!\left(I - V_k\,\mathrm{diag}(d_k)\,V_k^\top\right), + \qquad + d_i = \frac{\omega\sigma_i^2}{\omega\sigma_i^2 + \gamma}. +\end{equation} +% + +Applying this matrix to a vector $v$, the preconditioner is given via +% +\begin{equation} + M^{-1}v = \frac{1}{\gamma}\!\left(v - V_k^\top\bigl(d_k \odot (V_k v)\bigr)\right)~. +\end{equation} \subsection{CVXPY Reference Implementation} Before developing the Krylov solvers we establish a reference solution using CVXPY \cite{diamond2016}, a domain-specific -language for convex optimisation. The minimum variance problem maps directly onto a least-squares objective, which CVXPY +language for convex optimization. The minimum variance problem maps directly onto a least-squares objective, which CVXPY dispatches to a general-purpose interior-point solver. \begin{algorithm}[ht] \caption{CVXPY reference solver}\label{alg:cvxpy} \begin{algorithmic}[1] -\Require Returns matrix $X \in \mathbb{R}^{T \times n}$;\; shrinkage $\alpha$;\; return tilt $\rho\mu$;\; target $R$ +\Require Returns matrix $X \in \mathbb{R}^{T \times n}$;\; shrinkage $\alpha$;\; return tilt $\rho\hat \mu$; \Ensure Weight vector $w$;\; interior-point iteration count $k$ \State Declare $w \in \mathbb{R}^n$ as a CVXPY variable -\State Minimise $\tfrac{1-\alpha}{T}\|Xw\|^2 + \alpha\|Rw\|^2 - \rho\mu^\top w$ +\State Minimise $\tfrac{1}{T}\|Xw\|^2 - \rho\hat \mu^\top w$ \Statex\hspace{1.5em} subject to $\mathbf{1}^\top w = 1$,\; $w \geq \mathbf{0}$ \State Dispatch to Clarabel interior-point solver \cite{clarabel2024} \State \Return $w$,\; $k$ @@ -392,41 +497,42 @@ \subsection{CVXPY Reference Implementation} \end{algorithm} The formulation is transparent and correct, and CVXPY handles the long-only constraint natively. We use the Clarabel -interior-point solver \cite{clarabel2024}, which dispatches the problem as a second-order cone programme. Its cost is +interior-point solver \cite{clarabel2024}, which dispatches the problem as a second-order cone program. Its cost is substantial: interior-point methods require $O(n^3)$ work per iteration, and the overhead of problem construction and solver communication adds further latency. On the S\&P~500 problem ($n = 495$) this amounts to roughly 1.6 seconds (Table~\ref{tab:sp500}), making CVXPY unsuitable for production use but ideal as a ground-truth reference. -All three methods in this paper (CVXPY, KKT direct, and CG) solve the same optimisation problem and produce identical +All three methods in this paper (CVXPY, KKT direct, and CG) solve the same optimization problem and produce identical portfolio weights up to numerical tolerance. The comparison is purely computational: the question is not which method finds a better portfolio, but which finds the same portfolio fastest. The speedup figures in Table~\ref{tab:sp500} should be read with this in mind. CVXPY and -Clarabel are general-purpose tools: the same code handles additional convex constraints --- sector limits, turnover -bounds, factor exposure constraints --- without modification. The CG method exploits the specific structure of the pure +Clarabel are general-purpose tools: the same code handles additional convex constraints---sector limits, turnover bounds, factor-exposure constraints---without modification. The CG method exploits the specific structure of the pure long-only minimum variance problem and would require rework to accommodate constraints beyond the budget and non-negativity requirements. The speedup therefore conflates two effects: genuine algorithmic efficiency from exploiting problem structure, and the overhead of a framework designed for a far broader problem class. For the standard long-only constraint the full speedup is available; practitioners requiring richer constraint sets face a narrower tradeoff and -may find a specialised QP solver a more appropriate baseline. +may find a specialized QP solver a more appropriate baseline. \subsection{Direct Solve (Baseline)} -For small $n$, \texttt{numpy.linalg.solve} applied to the $n_a \times n_a$ covariance matrix is fast and exact. It -requires forming $X_{\mathrm{a}}^\top X_{\mathrm{a}}$ (an $O(n_a^2 T)$ operation), but for $n \ll T$ this is not the +For small $n$, \texttt{numpy.linalg.solve} applied to the $n_a \times n_a$ covariance matrix +\tcr{which covariance matrix, the sample covariance matrix or LW shrinkage?} +is fast and exact. \tcr{Should it not be exact always, also for large~$n$?} +It requires forming $X_{\mathrm{a}}^\top X_{\mathrm{a}}$ (an $O(n_a^2 T)$ operation), but for $n \ll T$ this is not the bottleneck. \begin{algorithm}[ht] \caption{Direct solver}\label{alg:kkt} \begin{algorithmic}[1] -\Require Problem data;\; regularisation target $R$ (optional) +\Require Problem data; \Ensure Weight vector $w$;\; outer elimination step count $s$ \Function{DirectSolve}{active} -\State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $R_{\mathrm{a}} \leftarrow R[\mathrm{active},\mathrm{active}]$,\; $n_a \leftarrow +\State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $n_a \leftarrow |\mathrm{active}|$ - \State Assemble $\Sigma_{\mathrm{a}} \leftarrow \tfrac{1-\alpha}{T}\,X_{\mathrm{a}}^\top X_{\mathrm{a}} + \alpha\,R_{\mathrm{a}}^\top R_{\mathrm{a}}$ + \State Assemble $\Sigma_{\mathrm{a}} \leftarrow \tfrac{1}{T}\,X_{\mathrm{a}}^\top X_{\mathrm{a}} \hfill\textit{(SPD, $n_a\times n_a$)} - \State Solve $\Sigma_{\mathrm{a}}\, v = \mathbf{1}$ by dense factorisation + \State Solve $\Sigma_{\mathrm{a}}\, v = \mathbf{1}$ by dense factorization \State \Return $v\,/\,(\mathbf{1}^\top v)$,\; $1$ \hfill\textit{(per-step count; no inner iterations)} \EndFunction \State $(w,\, s) \leftarrow \textsc{EliminationLoop}(\textsc{DirectSolve})$ @@ -438,14 +544,14 @@ \subsection{Direct Solve (Baseline)} \subsection{NNLS Solver}\label{ssec:nnls} - +\tcb{But NNLS is using an active-set method under the hood, too An alternative approach bypasses the primal-dual loop entirely by reformulating the long-only constraint as a non-negative least squares (NNLS) problem. Starting from the objective $\min_w \frac{1-\alpha}{T}\|Xw\|^2 + \alpha\|Rw\|^2$, the budget constraint $\mathbf{1}^\top w = 1$ is incorporated via a large-penalty row, yielding the augmented system: \[ - \min_{w \geq 0}\; \left\|\begin{pmatrix} \sqrt{(1-\alpha)/T}\,X \\ \sqrt{\alpha}\,R \\ M\,\mathbf{1}^\top \end{pmatrix} - w - \begin{pmatrix} \mathbf{0} \\ \mathbf{0} \\ M \end{pmatrix}\right\|^2, + \min_{w \geq 0}\; \left\|\begin{pmatrix} \sqrt{1/T}\,X \\ M\,\mathbf{1}^\top \end{pmatrix} + w - \begin{pmatrix} \mathbf{0} \\ M \end{pmatrix}\right\|^2, \] where $M = \|X\|_F \cdot T$ is a penalty large enough to enforce $\mathbf{1}^\top w \approx 1$ to machine precision. The non-negativity constraint $w \geq 0$ is handled natively by the Lawson-Hanson active-set algorithm @@ -457,8 +563,10 @@ \subsection{NNLS Solver}\label{ssec:nnls} algorithm implicitly forms normal equations $A^\top A$ where $A$ is the augmented matrix, incurring the same $O(n^2 T)$ cost as the direct KKT solve. With Ledoit-Wolf shrinkage, the augmented matrix grows from $T \times n$ to $(T+n) \times n$, making NNLS paradoxically slower with shrinkage than without, an inversion of the pattern seen for KKT and CG. -Table~\ref{tab:sp500} confirms this: on the S\&P~500 problem NNLS takes $0.038$\,s without -shrinkage but $0.086$\,s with it, while CG moves in the opposite direction. +Table~\ref{tab:sp500} confirms this: on the S\&P~500 problem, NNLS takes $0.038$\,s without +shrinkage but $0.086$\,s with it, whereas CG moves in the opposite direction. +} + \subsection{Proximal Gradient}\label{ssec:proximal} @@ -511,6 +619,7 @@ \subsection{Proximal Gradient}\label{ssec:proximal} \end{algorithmic} \end{algorithm} +\tbr{ \paragraph{Cost per iteration.} The matrix $\Sigma_{\mathrm{LW}}$ is assembled once at $O(n^2(T+n_R))$ cost (the same setup as the direct KKT solve) and cached; the largest eigenvalue $L$ is computed alongside it. Each @@ -553,64 +662,76 @@ \subsection{Proximal Gradient}\label{ssec:proximal} exploited further---for instance in the $\ell_1$-ball variant of the long-only constraint studied in \cite{duchi2008}---or when one prefers a conceptually simple, loop-free implementation at the cost of slower per-problem convergence. +} \section{Ledoit-Wolf Shrinkage} \label{sec:precond} -The sample covariance matrix $\frac{1}{T}X^\top X$ is a poor estimator when $n$ is large relative to $T$: its smallest -eigenvalues are severely downward-biased, and the eigenvalue spread (the ratio of largest to smallest) far exceeds that -of the true covariance. Ledoit and Wolf \cite{ledoit2004} propose replacing the sample covariance with a convex -combination of the sample estimate and a scaled identity: +It is well known that the sample covariance matrix $\hat \Sigma = \frac{1}{T}X^\top X$ is a poor estimator +of the true covariance matrix $\Sigma$ when $n$ is ``not small" relative to $T$: its smallest +eigenvalues are severely downward-biased, and the condition number (the ratio of largest to smallest eigenvalue) generally far exceeds that of the true covariance matrix. +Ledoit and Wolf \cite{ledoit2004} propose replacing the sample covariance matrix with a convex +combination of itself and the scaled identity matrix (called the {\em shrinkage target}): \[ -\Sigma_{\text{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\cdot\mathrm{target}, \quad \bar\lambda = \frac{\|X\|_F^2}{nT} +\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha\cdot \bar \lambda I \quad \mbox{with} \quad +\bar\lambda = \frac{\|X\|_{\text{F}}^2}{nT}~. \] -where $\bar\lambda$ is the average eigenvalue of the sample covariance and the classical choice is -$\mathrm{target} = \bar\lambda I$. The implementation uses a general $N\times N$ regularisation target matrix $R$ -(satisfying $\mathrm{target} = R^\top R$), so that -$\Sigma_{\text{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\,R^\top R$. +Here $\|\cdot\|_{\text{F}}$ denotes the Frobenius norm of a square matrix, and +$\bar\lambda$ thus denotes the average eigenvalue of the sample covariance matrix. +The implementation \tcr{Should we say "Our implementation"?} +uses a generalized $N\times N$ regularization shrinkage target matrix $R$ +(satisfying $\mathrm{target} = R^\top R$), \tcr{The target cannot satisfy both target = $R$ and +target = $R^\top R$} +so that +$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha \cdot R^\top R$. +\tcr{Maybe we need a more general notation here, to allow for a generalized shrinkage target.} Classical Ledoit-Wolf shrinkage is recovered by setting $R = \sqrt{\bar\lambda}\,I$; for i.i.d.\ $\mathcal{N}(0,1)$ synthetic returns $\bar\lambda \approx 1$ so $R = I$ is equivalent. The shrinkage intensity $\alpha \in (0,1)$ is -chosen analytically to minimise the expected Frobenius distance to the true covariance, requiring no cross-validation. +chosen analytically to minimize the expected Frobenius distance to the true covariance matrix, requiring no cross-validation. \tcr{This is not quite true. We are consistently estimating the optimal $\alpha$ only, we cannot guarantee to minimize the expected F-norm distance in finite samples.} The effect on the spectrum is direct: the target term lifts every eigenvalue, bounding the smallest away from zero and -compressing the overall spread. The condition number satisfies: +compressing the overall spread. \tcr{Not sure what is meant by ``lifts every eigenvalue". Classical LW shrinkage moves all sample eigenvalues towards their average. So large eigenvalues go down, whereas small eigenvalues go up.} +The condition number satisfies: \[ -\kappa(\Sigma_{\text{LW}}) = \frac{\lambda_{\max}(\Sigma_{\text{LW}})}{\lambda_{\min}(\Sigma_{\text{LW}})} \leq -\frac{(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)}{\alpha\lambda_{\min}(R^\top R)} +\kappa(\hat \Sigma_{\text{LW}}) = \frac{\lambda_{\max}(\hat \Sigma_{\text{LW}})}{\lambda_{\min}(\hat \Sigma_{\text{LW}})} \leq +\frac{(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)}{\alpha\lambda_{\min}(R^\top R)}~, \] -since $\lambda_{\min}(\Sigma_{\text{LW}}) \geq \alpha\lambda_{\min}(R^\top R)$ and, by Weyl's inequality applied to -$\Sigma_{\text{LW}} = (1-\alpha)\hat\Sigma + \alpha R^\top R$, $\lambda_{\max}(\Sigma_{\text{LW}}) \leq +since $\lambda_{\min}(\hat \Sigma_{\text{LW}}) \geq \alpha\lambda_{\min}(R^\top R)$ and, by Weyl's inequality applied to +$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot \hat\Sigma + \alpha \cdot R^\top R$, $\lambda_{\max}(\hat \Sigma_{\text{LW}}) \leq (1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)$, where $\lambda_{\max}(\hat\Sigma) = \lambda_{\max}(X^\top X/T)$ is -the largest sample eigenvalue. As $\alpha \to 1$ the bound approaches $\lambda_{\max}(R^\top R)/\lambda_{\min}(R^\top R)$, +the largest sample eigenvalue. As $\alpha \to 1$, the bound approaches $\lambda_{\max}(R^\top R)/\lambda_{\min}(R^\top R)$, which equals $1$ for the classical choice $R = \sqrt{\bar\lambda}\,I$, correctly reflecting that full shrinkage yields a scalar multiple of the identity. For fixed data, the condition number decreases monotonically as $\alpha$ increases, with the rate governed by the spectral ratio $\lambda_{\max}(\hat\Sigma)/\lambda_{\min}(R^\top R)$. For CG applied to -$\Sigma_{\text{LW}}\,v = \mathbf{1}$, the convergence rate is governed by $\kappa(\Sigma_{\text{LW}})$; compressing the +$\hat \Sigma_{\text{LW}}\,v = \mathbf{1}$, the convergence rate is governed by $\kappa(\hat \Sigma_{\text{LW}})$; compressing the spectrum directly reduces iteration counts. Subsequent work by the same authors provides an analytical nonlinear -shrinkage formula \cite{ledoit2020} that applies a different coefficient to each sample eigenvalue, achieving a smaller -Frobenius loss when $n/T$ is not negligible. Nonlinear shrinkage cannot be expressed as a simple augmented matrix +shrinkage scheme that applies a different coefficient to each sample eigenvalue, typically achieving a smaller +Frobenius loss than linear shrinkage when $n/T$ is not negligible; see \cite{ledoit2020}. +Unfortunately, nonlinear shrinkage cannot be expressed as a simple augmented matrix $\tilde{X}$, so incorporating it into the matrix-free framework would require a separate Krylov-compatible preconditioner rather than a modified return matrix. -In the implementation, shrinkage is applied by augmenting the return matrix rather than forming $\Sigma_{\text{LW}}$ +In the implementation, shrinkage is applied by augmenting the return matrix rather than forming $\hat \Sigma_{\text{LW}}$ explicitly. Setting $c = 1 - \alpha$, the stacked matrix $\tilde{X} = [\sqrt{c/T}\,X;\,\sqrt{\alpha}\,R]$ satisfies $\tilde{X}^\top \tilde{X} = \frac{c}{T}\,X^\top X + \alpha\,R^\top R -= c\,\hat\Sigma + \alpha\,R^\top R = \Sigma_{\text{LW}}$, += c\,\hat\Sigma + \alpha\,R^\top R = \hat \Sigma_{\text{LW}}$, so any of the solvers above can be applied to $\tilde{X}$ and produce exactly the LW-shrinkage portfolio weights. -For $R = \sqrt{\bar\lambda}\,I$ this gives classical Ledoit-Wolf: $\Sigma_{\text{LW}} = (1-\alpha)\,\hat\Sigma + \alpha\bar\lambda\,I$. +For $R = \sqrt{\bar\lambda}\,I$, this results in the classical Ledoit-Wolf linear shrinkage: $\hat \Sigma_{\text{LW}} = (1-\alpha) \cdot \hat\Sigma + \alpha \cdot \bar\lambda\,I$. From the perspective of numerical linear algebra, compressing the eigenvalue spectrum of the system matrix is precisely what a preconditioner does. Ledoit-Wolf shrinkage therefore plays a dual role: it reduces statistical estimation error -and, as a side effect, accelerates iterative solvers by clustering the eigenvalues. The two motivations are independent -but arrive at the same modified matrix. +and, as a side effect, accelerates iterative solvers by reducing the condition number of the input matrix relative the sample covariance matrix. The two motivations are independent +but arrive at the same modified matrix. \tcr{I do not quite understand the meaning of this last sentence.} -The proportional identity shrinkage target $\alpha\bar\lambda I$ is the simplest choice, but the preconditioning -literature suggests sharper alternatives. A factor-model preconditioner, replacing $\alpha\bar\lambda I$ with a +The proportional identity shrinkage target $\alpha \cdot \bar\lambda I$ is the simplest choice, but the preconditioning +literature suggests sharper alternatives. A factor-model preconditioner, replacing $\alpha \cdot \bar\lambda I$ with a low-rank-plus-diagonal approximation $BB^\top + D$, would cluster eigenvalues more tightly at the cost of a slightly -more expensive application. Incomplete Cholesky factorisation of the shrunk matrix provides a general-purpose option. -For rolling-window problems where the covariance changes slowly between periods, a preconditioner built from the -previous period's factorisation could be reused with minimal update cost. None of these require forming the covariance +more expensive application. Incomplete Cholesky factorization of the shrunk matrix provides a general-purpose option. +For rolling-window problems where the estimated covariance matrix changes slowly between periods, a preconditioner built from the +previous period's factorization could be reused with minimal update cost. None of these +\tcr{none of these what, approaches, methods, \ldots?} +require forming the covariance matrix explicitly and each is compatible with the matrix-free framework described above. \section{Numerical Results}\label{sec:results} @@ -618,7 +739,7 @@ \section{Numerical Results}\label{sec:results} All experiments were run on an Apple M4 Pro (14-core CPU, 48\,GB RAM) under Python~3.12, NumPy~2.4, SciPy~1.17, CVXPY~1.8.2, and Clarabel~0.11.1. The iterative solvers use SciPy's default relative residual tolerance of $10^{-5}$, which was reached in all reported runs; reported times are the minimum of three repeated solves and cover only the -solve itself --- construction of the shrinkage target $\hat\sigma^2 I$ is excluded, as it requires a single pass over +solve itself---construction of the shrinkage target $\hat\sigma^2 I$ is excluded, as it requires a single pass over $X$ and is negligible relative to any solver. All implementations are available in the companion package \cite{fast_minimum_variance}. @@ -627,34 +748,35 @@ \subsection*{S\&P~500 benchmark} The dataset consists of daily percentage returns for 495 S\&P~500 constituents over the period July~2021--May~2026 (1192 trading days, $T/n \approx 2.4$), sourced from Yahoo Finance via the \texttt{yfinance} Python package. -Tickers with more than 5\% missing observations are excluded; short gaps are forward-filled. The return +Tickers with more than 5\% missing observations are excluded; short gaps are forward-filled. \tcr{Please explain what is meant by ``forward-filled"? Why not replace missing values by zero?} +The return matrix $X$ is formed by demeaning the returns column-wise. -\paragraph{Spectrum of $\hat\Sigma = \tfrac{1}{T}X^\top X$.} +\paragraph{Spectrum of $\hat\Sigma = \tfrac{1}{T}X^\top X$} The eigenvalue structure of the sample covariance is qualitatively different from that of a random Gaussian matrix and is the reason the S\&P~500 problem is a meaningful test case for the preconditioning argument. For a random $T \times n$ matrix with i.i.d.\ $\mathcal{N}(0,\sigma^2)$ entries, the Marchenko-Pastur law predicts that the bulk of the spectrum lies in the interval $[\sigma^2(1-\sqrt{n/T})^2,\,\sigma^2(1+\sqrt{n/T})^2]$, giving a mild condition number that grows modestly with $n/T$. The equity spectrum is nothing like this. -The largest eigenvalue is $\lambda_1 \approx 0.067$, the market factor, which alone accounts for $\approx 30\%$ of -total portfolio variance. To apply the Marchenko-Pastur law we estimate $\sigma^2$ as the average squared entry of $X$, +The largest eigenvalue is $\lambda_1 \approx 0.067$, corresponding the market factor, which alone accounts for $\approx 30\%$ of +total portfolio variance. To apply the Marchenko-Pastur law, we estimate $\sigma^2$ as the average squared entry of $X$, \[ - \hat\sigma^2 = \frac{\|X\|_F^2}{nT} \approx 0.000457, + \hat\sigma^2 = \frac{\|X\|_F^2}{nT} \approx 0.000457~, \] the natural entry-level variance of a daily percentage return. The MP upper edge $\hat\sigma^2(1+\sqrt{n/T})^2 \approx 0.00123$ then identifies 22 eigenvalues as statistically significant factors; the remaining 473 are consistent with noise yet still span a wide range down to $\lambda_{\min} \approx 6 \times 10^{-7}$. The resulting condition number is \[ - \kappa(\hat\Sigma) = \frac{\lambda_{\max}}{\lambda_{\min}} \approx 109{,}000, + \kappa(\hat\Sigma) = \frac{\lambda_{\max}}{\lambda_{\min}} \approx 109{,}000~, \] far larger than the $O(10)$--$O(100)$ range typical of random matrices at this aspect ratio. The first ten eigenvalues account for half the total variance, while the bottom half of the spectrum contributes less than $3\%$. This extreme imbalance---a handful of dominant factors and a long tail of near-zero noise eigenvalues---is precisely what makes CG slow without preconditioning (744 iterations) and what makes shrinkage decisive: lifting -$\lambda_{\min}$ by even a moderate $\alpha$ collapses the condition number and cuts the iteration count to 82. +$\lambda_{\min}$ by even a moderate $\alpha$ dramatically reduces the condition number and thus the iteration count to 82. -Table~\ref{tab:sp500} reports timings and iteration counts for all solvers on this dataset. +Table~\ref{tab:sp500} reports timings and iteration counts on this dataset for the various solvers we consider. \begin{table}[ht] \centering @@ -685,27 +807,28 @@ \subsection*{S\&P~500 benchmark} \end{table} The real-data results reveal two effects. First, without shrinkage the equity system is hard to solve iteratively: -CG requires 744 iterations and is slower than KKT direct (0.0120\,s vs.\ 0.0056\,s). The strong factor structure of +CG requires 744 iterations and is slower than KKT direct (0.0120\,s vs 0.0056\,s). The strong factor structure of equity returns concentrates variance along a few eigendirections, inflating the condition number and offsetting the matrix-free advantage of CG. -Second, Ledoit-Wolf shrinkage benefits both iterative and direct solvers: it cuts CG iterations from 744 to~82 and +Second, Ledoit-Wolf shrinkage benefits both iterative and direct solvers: it reduces CG iterations from 744 to~82 and reduces active-set outer steps from~22 to~6, since a lower condition number means fewer assets end up with negative unconstrained weights and the primal-dual loop converges faster. At $n=495$, KKT direct (0.0062\,s, $399\times$ over CVXPY) and CG (0.0042\,s, $594\times$) are both well under a millisecond and within a factor of~1.5 of each other. -For portfolios of this size KKT direct is the practical recommendation: it requires no convergence tolerance and is +For portfolios of this size, KKT direct is the practical recommendation: it requires no convergence tolerance and is faster without shrinkage. CG's advantage lies at larger $n$, where avoiding explicit assembly of $\tilde{X}^\top\tilde{X}$ reduces per-step cost from $O(n^2)$ to $O(Tn)$ (Figure~\ref{fig:scaling}). The ``Clarabel (direct API)'' row confirms the speedup is not merely framework overhead: bypassing CVXPY reduces -Clarabel's time from 2.5\,s to 0.092\,s, yet KKT direct ($0.006$\,s) is still $15\times$ faster. These results are -specific to this five-year window; iteration counts depend on the factor structure of equity returns and may differ +Clarabel's time from 2.5\,s to 0.092\,s, yet KKT direct ($0.006$\,s) is still $15\times$ faster. Not that these results may be +specific to five-year window considered; iteration counts depend on the factor structure of equity returns and may differ in other market regimes. \subsection*{Scaling with problem size} Figure~\ref{fig:scaling} uses synthetic i.i.d.\ Gaussian data to isolate the computational scaling of each solver across a range of problem sizes and shrinkage levels; the spectral properties of the data are irrelevant here since the -point is to measure cost growth, not iteration counts. The left panel shows wall-clock time as a function of $n$ (with +point is to measure cost growth, not iteration counts. +The left panel shows wall-clock time as a function of $n$ (with $T = 2n$ and LW shrinkage throughout). The KKT direct solve scales as $O(n^3)$, dominated by the $n \times n$ linear system. Each Krylov iteration requires $O(nT) = O(n^2)$ work for the two matrix-vector products with $X$; with the spectrum compressed by shrinkage the iteration count grows slowly @@ -714,7 +837,7 @@ \subsection*{Scaling with problem size} The right panel uses the rank-deficient setting $T = 250 < n = 500$, in which $X^\top X$ has at most $T$ non-zero eigenvalues and the unshrunk sample covariance is singular. This regime is practically relevant: asset universes with more instruments than trading days arise when securities with limited common history are pooled, or when a large -cross-section is combined with a short lookback window. Without shrinkage the KKT system is singular and the iterative +cross-section is combined with a short lookback window. Without shrinkage, the KKT system is singular and the iterative solvers do not converge; every point in the panel has $\alpha > 0$. The preconditioning effect of shrinkage is strongest precisely here: iteration counts fall steeply as $\alpha$ increases from zero, reaching the practical operating point at moderate $\alpha$ values well below unity. @@ -722,17 +845,17 @@ \subsection*{Scaling with problem size} \begin{figure}[ht] \centering \includegraphics[width=\linewidth]{graphs/minvar_scaling.pdf} -\caption{Left: wall-clock time vs.\ number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied -throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). Right: CG iteration counts vs.\ shrinkage -intensity $\alpha$ at fixed $n=500$, $T=250$ (rank-deficient baseline). Larger $\alpha$ compresses the eigenvalue -spectrum and reduces the condition number $\kappa(\Sigma_{\mathrm{LW}})$, directly reducing CG iterations.} +\caption{Left: wall-clock time vs number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied +throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). Right: CG iteration counts vs shrinkage +intensity $\alpha$ at fixed $n=500$, $T=250$ (rank-deficient baseline). Larger values of $\alpha$ compress the eigenvalue +spectrum and reduce the condition number $\kappa(\hat \Sigma_{\mathrm{LW}})$, thereby reducing CG iterations.} \label{fig:scaling} \end{figure} \begin{figure}[ht] \centering \includegraphics[width=0.65\linewidth]{graphs/minvar_loglog.pdf} -\caption{Wall-clock time vs.\ number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied +\caption{Wall-clock time vs number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). KKT direct scales as $O(n^3)$; matrix-free CG scales empirically as $O(n^2)$ due to the iteration count growing slowly with the compressed spectrum; NNLS also scales as $O(n^3)$ (Lawson--Hanson normal equations). Reference slopes are anchored to the KKT curve at $n = 500$.} @@ -741,34 +864,39 @@ \subsection*{Scaling with problem size} \section{Conclusion} -The long-only minimum variance portfolio problem, formulated directly in terms of the returns matrix, -reduces to solving the $n\times n$ symmetric positive definite system $\Sigma_{\mathrm{LW}}\, v = \mathbf{1}$ via +The long-only minimum variance portfolio problem, formulated +directly in terms of the returns matrix, +reduces to solving the $n\times n$ symmetric positive definite system $\hat \Sigma_{\mathrm{LW}}\, v = \mathbf{1}$ via conjugate gradients, followed by a single rescaling $w = v/(\mathbf{1}^\top v)$. The covariance matrix is never formed; all operations are matrix-vector products with $X$. +\tcr{This paragraph has to be reworked. What is meant by ``formulated directly in terms of the returns matrix"? +This sounds like the input is the sample covariance matrix. But then the input is the LW matrix, which requires more than just the returns matrix. Also, a conclusion has to be somewhat self-contained, so one cannot just throw out terms like CVXPY or KKT.} The empirical results on 495 S\&P~500 constituents over five years confirm the practical value of this approach. For portfolios up to $n \approx 1000$, KKT direct is already 300--400$\times$ faster than CVXPY across both shrinkage regimes and requires no convergence tolerance: it is the practical recommendation. Without shrinkage, the matrix-free advantage of CG is offset by high iteration counts (744 on S\&P~500 equity data) driven by the strong factor structure of equity returns. Ledoit-Wolf shrinkage \cite{ledoit2004} benefits both solvers: it compresses -the eigenvalue spectrum, cuts CG iterations to~82, and reduces active-set outer steps from~22 to~6, since a lower +the eigenvalue spectrum, reduces CG iterations to~82, and reduces active-set outer steps from~22 to~6, since a lower condition number shrinks the unconstrained minimum-variance portfolio toward equal weights and fewer assets receive -negative weights in the primal step. At $n=495$ CG (0.0042\,s) edges KKT direct (0.0062\,s) by a factor of~1.5. -CG's advantage becomes decisive at large $n$: the scaling experiments show CG is roughly $2\times$ faster than KKT -at $n=2000$, a gap that widens further as $n$ grows. +negative weights in the primal step. \tcr{That sentence is really a bit too long.} +At $n=495$ CG (0.0042\,s) edges KKT direct (0.0062\,s) by a factor of~1.5. +CG's advantage becomes decisive at large $n$: the scaling experiments show CG is roughly two times faster than KKT +at $n=2000$, a gap that widens even further as $n$ grows. The shrinkage estimator of Ledoit and Wolf and the preconditioning theory of Golub and Van Loan \cite{golub2013} address -different problems but act on the same object: the eigenvalue spectrum of the system matrix. This connection, which the +different problems but act on the same object, the eigenvalue spectrum of the system matrix. This connection, which the two literatures do not usually make explicit, points toward richer preconditioners (factor-model or low-rank structured) as a productive direction for future work. -The SPD structure extends naturally beyond the minimum variance setting. Adding a return term $\rho\mu^\top w$ to the -objective requires solving two systems ($\Sigma_{\mathrm{LW}}\, v_1 = \mathbf{1}$ and $\Sigma_{\mathrm{LW}}\, v_2 = \mu$) and recovering the budget -multiplier analytically; the full Markowitz mean-variance problem is handled in a companion paper. +The SPD structure extends naturally beyond the minimum variance setting. Adding a return term $\rho\hat \mu^\top w$ to the +objective requires solving two systems ($\hat \Sigma_{\mathrm{LW}}\, v_1 = \mathbf{1}$ and $\hat \Sigma_{\mathrm{LW}}\, v_2 = \hat \mu$) and recovering the budget +multiplier analytically. The full Markowitz mean-variance problem will be addressed in future work in a companion paper. -\bibliographystyle{siam/siamplain} +% \bibliographystyle{siam/siamplain} +\bibliographystyle{siam} \bibliography{bib/refs} \end{document} From e5da546b215c70d3b9ac44f67f815a131fe877b1 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:42:11 +0400 Subject: [PATCH 09/39] docs: add closing sentence to Section 3 stating the SPD + inequality handoff Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 3 +++ 1 file changed, 3 insertions(+) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 6f3ad6a..b2e58df 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -316,6 +316,9 @@ \section{Positive-Definite Reduction}\label{sec:pd} and the weights follow as $w = \tfrac{\lambda}{2}\,v_1 + \tfrac{\rho}{2}\,v_2$. In both cases the $(n+1)\times(n+1)$ saddle-point system is never needed and $\lambda$ is recovered analytically rather than as part of the solve. +The long-only problem therefore reduces to solving a symmetric positive definite system +$\hat\Sigma_{\mathrm{LW}}\,v = b$ for one or two right-hand sides, subject to the remaining +inequality constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}. \section{Primal-Dual Loop}\label{sec:primaldual} From 7b39fdb7ad6eb69d1c721952b9ac3be1b3ebbec7 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:49:16 +0400 Subject: [PATCH 10/39] feat: matrix-free proximal gradient via power iteration Replace explicit Sigma_LW assembly with matrix-free gradient mat.T @ (mat @ w) and estimate the Lipschitz constant via power iteration on mat, both O(nT) per step. Co-Authored-By: Claude Sonnet 4.6 --- src/fast_minimum_variance/_base.py | 49 --------------------------- src/fast_minimum_variance/proximal.py | 44 ++++++++++++++++++++---- 2 files changed, 37 insertions(+), 56 deletions(-) diff --git a/src/fast_minimum_variance/_base.py b/src/fast_minimum_variance/_base.py index 4d77883..b39c9b6 100644 --- a/src/fast_minimum_variance/_base.py +++ b/src/fast_minimum_variance/_base.py @@ -327,52 +327,3 @@ def solve_proximal(self, *, project: bool = True): w = self._clip_and_renormalize(w) return w, 1 - def solve_osqp(self, *, project: bool = True): - """Solve via OSQP (operator-splitting QP solver, direct API, no CVXPY overhead). - - Assembles ``P = 2·Σ_LW`` as a sparse upper-triangular CSC matrix and - calls OSQP directly. The problem-specific constraint data is supplied - by ``_osqp_constraints``. Returns ``(w, iters)`` where ``iters`` is - the number of ADMM iterations. - - Args: - project: Clip and renormalize after solving (see ``solve_kkt``). - - Returns: - ``(w, n_iters)`` — weight vector of shape ``(N,)`` and number of - ADMM iterations. - - Examples: - >>> import numpy as np - >>> from fast_minimum_variance import Problem - >>> X = np.random.default_rng(0).standard_normal((100, 5)) - >>> w, iters = Problem(X).solve_osqp() - >>> float(round(w.sum(), 6)) - 1.0 - >>> bool((w >= -1e-6).all()) - True - """ - n = self.n - - if self.target is None: - p_dense = 2.0 * ((self.X.T @ self.X) / self.t) - else: - p_dense = 2.0 * ((1 - self.alpha) * (self.X.T @ self.X) / self.t + self.alpha * self.target) - - # p_dense = 2.0 * ((1-self.alpha) * (self.X.T @ self.X)/self.t + self.alpha * self.target) - p_upper = triu(p_dense, format="csc") - - q = np.zeros(n) - if self.rho != 0.0 and self.mu is not None: - q = -self.rho * self.mu - - a_mat, l_vec, u_vec = self._osqp_constraints() - - prob = osqp.OSQP() - prob.setup(p_upper, q, a_mat, l_vec, u_vec, verbose=False) - res = prob.solve() - - w = np.array(res.x) - if project: - w = self._clip_and_renormalize(w) - return w, res.info.iter diff --git a/src/fast_minimum_variance/proximal.py b/src/fast_minimum_variance/proximal.py index 3fdac41..9910ce9 100644 --- a/src/fast_minimum_variance/proximal.py +++ b/src/fast_minimum_variance/proximal.py @@ -4,6 +4,10 @@ optimization on the probability simplex. The algorithm is based on iterative projection using the efficient simplex projection from Duchi et al. (2008). +The gradient is evaluated matrix-free as mat.T @ (mat @ w), avoiding explicit assembly +of the n x n normal matrix mat.T @ mat. The Lipschitz constant is estimated via power +iteration, also matrix-free. + References: ---------- @@ -72,6 +76,30 @@ def proj_simplex( return result +def _lipschitz( + mat: NDArray[np.floating], + n_iter: int = 30, + rng: np.random.Generator | None = None, +) -> float: + """Estimate lambda_max(mat.T @ mat) via power iteration (matrix-free). + + Each iteration costs O(rows * cols) — two matrix-vector products with mat — + and never forms the cols x cols normal matrix. + """ + if rng is None: + rng = np.random.default_rng() + v = rng.standard_normal(mat.shape[1]) + v /= np.linalg.norm(v) + lip = 1.0 + for _ in range(n_iter): + w = mat.T @ (mat @ v) + lip = float(np.linalg.norm(w)) + if lip < 1e-15: + return lip + v = w / lip + return lip + + def prox_gradient( mat: NDArray[np.floating], vec: NDArray[np.floating], @@ -84,9 +112,10 @@ def prox_gradient( minimize 0.5 ||mat @ x - vec||^2 subject to x >= 0, sum(x) = 1 - The function uses proximal gradient descent with simplex projection to find - the solution. The step size is determined by the Lipschitz constant of the - gradient. + The gradient mat.T @ (mat @ x) - mat.T @ vec is evaluated matrix-free at each + step; the normal matrix mat.T @ mat is never formed. The Lipschitz constant + L = lambda_max(mat.T @ mat) is estimated once via power iteration, also + matrix-free, at O(n_power_iter * rows * cols) setup cost. Parameters ---------- @@ -118,15 +147,16 @@ def prox_gradient( """ rng = np.random.default_rng() prim_var: NDArray[np.floating] = np.asarray(rng.standard_normal(size=mat.shape[1])) - sym_mat = mat.T @ mat - lip = np.linalg.norm(sym_mat, 2) # Lipschitz constant of the gradient - step = 0.5 / lip if abs(lip) > 1e-15 else 1.0 + lip = _lipschitz(mat, rng=rng) + step = 0.5 / lip if lip > 1e-15 else 1.0 + # Precompute mat.T @ vec once; zero for minimum-variance (vec = 0). out_prod = mat.T @ vec ite = 0 err_rel = eps_rel + 1 while err_rel > eps_rel and ite < max_iter: - prim_var_new = proj_simplex(prim_var - step * (sym_mat @ prim_var - out_prod)) + grad = mat.T @ (mat @ prim_var) - out_prod + prim_var_new = proj_simplex(prim_var - step * grad) err_rel = float(np.linalg.norm(prim_var - prim_var_new, 2)) prim_var = prim_var_new.copy() ite += 1 From 355335177d842e483a08ae6c7f7118fa16a98f8a Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 18:55:42 +0400 Subject: [PATCH 11/39] docs: rewrite Section 4 to introduce both active-set and simplex-projection strategies Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 151 ++++++++++++++++------------------------- 1 file changed, 57 insertions(+), 94 deletions(-) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index b2e58df..55f85c1 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -232,62 +232,6 @@ \section{Problem Formulation}\label{sec:problem} both hold, which---together with stationarity from each inner solve---is sufficient for global optimality. -%\subsection*{Problem variants} -% -%The framework handles a range of standard portfolio construction %problems directly. Each is expressed as -%\begin{equation} -% \min_{w}\;\|\tilde{X}w\|^2 - \rho\mu^\top w -% \quad\text{subject to}\quad -% \mathbf{1}^\top w = 1,\;\; w \geq 0~, -% \label{eq:general} -%\end{equation} -%through choices of $\tilde{X}$, $\rho$, and $\mu$ summarized in %Table~\ref{tab:problems}. No change in the -%algorithm is required; only the data passed to the solver change. - -%\begin{table}[ht] -%\centering -%\begin{tabular}{llll} -%\toprule -%Portfolio problem & $\tilde{X}$ & $\rho$ & $\mu$ \\ -%\midrule -%Minimum variance & $X$ & $0$ & --- \\ -%Mean-variance (Markowitz) & $X$ & $>0$ & expected returns \\ -%Minimum tracking error$^*$ & $X$ & $2$ & $X^\top(Xb)$ \\ -%LW-regularized minimum variance & $[\sqrt{(1-%\alpha)/T}\,X;\,\sqrt{\alpha}\,R]$ & $0$ & --- \\ -%Factor-model covariance $FF^\top+D$ & $[F;\,D^{1/2}]$ & $0$ & --- \\ -%Regime-blended / multi-period & $[\omega_1 X_1;\,\ldots;\,\omega_K %X_K]$ & $0$ & --- \\ -%\bottomrule -%\end{tabular} -%\caption{Portfolio construction problems solved by the same primal--dual loop. -% $b$ is the benchmark weight vector; $F$ contains factor returns; $D$ is a diagonal specific-risk matrix; -% $R$ is the $N\times N$ regularisation target matrix (e.g.\ $\sqrt{\bar\lambda}\,I$ for classical Ledoit-Wolf -% shrinkage, where $\bar\lambda = \|X\|_F^2/(nT)$ is the mean eigenvalue of $\hat\Sigma$). -% Stacking of $\tilde{X}$ is in the row direction. -% $^*\min_w (w-b)^\top\Sigma(w-b) = \min_w w^\top\Sigma w - 2(\Sigma b)^\top w + \mathrm{const}$, so -% $\rho\mu = 2\Sigma b = 2X^\top(Xb)$ (two matrix--vector products). -% Since $\mu = \Sigma b$, the second SPD solve yields $v_2 = b$ exactly; tracking error adds no inner-solve cost.} -%\label{tab:problems} -%\end{table} - -%\paragraph{What does not fit} -%The maximum Sharpe ratio portfolio, also called the tangency portfolio, solves -%\[ -%\arg\max_w\; \frac{\mu^\top w}{\sqrt{w^\top\Sigma w}} -%\quad\text{subject to}\quad \mathbf{1}^\top w = 1,\; w\geq 0. -%\] -%This is a fractional programme and does not reduce to \eqref{eq:general} with a fixed $\rho$. -%It can be -%recovered parametrically by sweeping $\rho$ over the efficient frontier and locating the tangency point, but this -%requires multiple solves. Problems with additional linear inequality constraints---sector limits, turnover -%bounds, factor-exposure limits---also lie outside the current template; a general QP solver (e.g.\ CVXPY with -%Clarabel) is the appropriate tool for those. - -%The one structural requirement is that the covariance admit a factorization $\Sigma = \tilde{X}^\top\tilde{X}$ -%with $\tilde{X}$ available directly or by cheap row-augmentation, so that the matrix-vector product -%$v\mapsto\tilde{X}^\top(\tilde{X}v)$ remains $O(T'n)$ per iteration. Nonlinear shrinkage~\cite{ledoit2020}, -%which applies a separate coefficient to each sample eigenvalue, does not admit such a factorization and requires -%a different preconditioning strategy. - \section{Positive-Definite Reduction}\label{sec:pd} Ignoring the inequality constraints for now, the Lagrangian for the equality-constrained problem is: @@ -318,51 +262,41 @@ \section{Positive-Definite Reduction}\label{sec:pd} recovered analytically rather than as part of the solve. The long-only problem therefore reduces to solving a symmetric positive definite system $\hat\Sigma_{\mathrm{LW}}\,v = b$ for one or two right-hand sides, subject to the remaining -inequality constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}. +inequality constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}, +which presents two strategies for doing so. -\section{Primal-Dual Loop}\label{sec:primaldual} -The long-only constraint $w \geq 0$ is enforced by a \emph{primal--dual iterative procedure}. -Each outer step solves the equality-constrained (budget-only) $n_a \times n_a$ positive -definite system over the current active set. In the \emph{primal step}, any asset whose weight -is negative is dropped. Once all active weights are non-negative, a \emph{dual step} checks -the KKT gradient condition for every excluded asset: for global optimality, every excluded -asset $i$ must satisfy -\[ -\nabla_i f(w) \;\geq\; \lambda\;, -\quad\text{and}\quad -\nabla_i f(w) = \tfrac{2}{T}(X^\top X w)_i - \rho\hat\mu_i\;, -\] -where $\lambda$ is the budget Lagrange multiplier. An excluded asset that fails this condition -would improve the objective if added at a small positive level; it is re-added to the active -set and the loop repeats. The loop terminates when both primal and dual feasibility hold -simultaneously; combined with stationarity from the inner solve, this is sufficient for -global optimality. +\section{Enforcing Non-Negativity}\label{sec:primaldual} +Section~\ref{sec:pd} reduces the long-only problem to solving an SPD system, leaving +$w \geq 0$ as the sole remaining constraint. Two strategies are available. +\paragraph{Active-set strategy.} +The first strategy separates the budget constraint from the non-negativity constraint. +The budget-only problem is solved over a dynamically adjusted \emph{active set} of +assets; non-negativity is then enforced iteratively without ever modifying the inner +solve. -\tcb{This comes all a bit too early: -With Ledoit-Wolf shrinkage, the asset elimination loop converges rapidly in practice. Let $\bar\lambda = \|X\|_F^2/(nT)$ -denote the average eigenvalue of the sample covariance. The unconstrained minimizer satisfies $w^\star \propto -\hat \Sigma_{\mathrm{LW}}^{-1}\mathbf{1}$. As $\alpha \to 1$, $\hat \Sigma_{\mathrm{LW}} \to \bar\lambda I$ and $w^\star \to -\frac{1}{n}\mathbf{1}$, the equal-weighted $1/n$ portfolio with all components strictly positive. By continuity, there exists a -threshold $\alpha^\star \in (0,1)$ above which every component of the unconstrained solution is positive; for $\alpha > -\alpha^\star$ the first solve requires no asset to be dropped. The first-order expansion +In the \emph{primal step}, any asset with a negative weight is dropped from the active +set and the budget-constrained SPD system is re-solved on the reduced universe. Once +all active weights are non-negative, the \emph{dual step} checks the KKT gradient +condition for every excluded asset: global optimality requires every excluded asset $i$ +to satisfy \[ -\hat \Sigma_{\mathrm{LW}}^{-1}\mathbf{1} \approx \tfrac{1}{\alpha\bar\lambda}\mathbf{1} - -\tfrac{1-\alpha}{\alpha^2\bar\lambda^2 T}X^\top X\mathbf{1}~, + \nabla_i f(w) \;\geq\; \lambda\;, + \qquad + \nabla_i f(w) = \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i\;, \] -valid for $\alpha$ close to one, shows that $\alpha^\star$ is governed by the spectral structure of $X^\top X$: assets -with large total covariance exposure are the last to turn positive as $\alpha$ increases. For the LW-optimal $\alpha$ \tcr{A brief explanation would be needed here, optimal in what sense?} -and dense equity returns, the threshold is typically exceeded \tcr{This is an empirical finding, I guess? There cannot be an theory for such a finding} and the first solve yields few negative components. Across -50 experiments with $n$ from 100 to 1000 and $T = 2n$, the elimination loop required 2--3 outer iterations in all cases; -runtime is dominated by the linear solves. \tcr{Some reference to such experiments would be useful, that statement comes a bit ``out of the blue", without underlying details.} -} +where $\lambda$ is the budget Lagrange multiplier recovered analytically from +Section~\ref{sec:pd}. An excluded asset that violates this condition would decrease +portfolio variance if held at a small positive weight; it is re-added to the active set +and the loop repeats. The loop terminates when both primal and dual feasibility hold; +together with stationarity from the inner solve this certifies global optimality. \begin{algorithm}[ht] -\caption{Primal-dual outer loop}\label{alg:elim} +\caption{Active-set outer loop (wraps any inner solver $f$)}\label{alg:elim} \begin{algorithmic}[1] -\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$; +\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$ \Ensure Weight vector $w \in \mathbb{R}^n$;\; total inner iteration count $k$ \State $\mathrm{active} \leftarrow \{1,\ldots,n\}$;\; $k \leftarrow 0$ \Loop @@ -372,11 +306,11 @@ \section{Primal-Dual Loop}\label{sec:primaldual} \hfill\textit{(primal step: drop violators)} \Else \State $w_i \leftarrow w_{a,i}$ for $i \in \mathrm{active}$;\; $w_i \leftarrow 0$ for $i \notin \mathrm{active}$ - \State $g_i \leftarrow \tfrac{2}{T}(X^\top X w)_i - \rho\hat \mu_i$ + \State $g_i \leftarrow \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i$ \State $\lambda \leftarrow \mathrm{mean}_{j \in \mathrm{active}}(g_j)$ \State $\mathcal{V} \leftarrow \{i \notin \mathrm{active} : g_i < \lambda - \varepsilon\}$ \If{$\mathcal{V} = \emptyset$} - \State \textbf{break} \hfill\textit{(KKT conditions satisfied; globally optimal)} + \State \textbf{break} \hfill\textit{(KKT satisfied; globally optimal)} \EndIf \State $\mathrm{active} \leftarrow \mathrm{active} \cup \mathcal{V}$ \hfill\textit{(dual step: re-add violators)} @@ -386,7 +320,36 @@ \section{Primal-Dual Loop}\label{sec:primaldual} \end{algorithmic} \end{algorithm} -Algorithm~\ref{alg:elim} accepts any inner solver $f$; Section~\ref{sec:solvers} presents concrete choices for $f$. +\paragraph{Simplex-projection strategy.} +The second strategy treats the two constraints as a single geometric object: the +probability simplex +$\Delta_n = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 1,\, w \geq 0\}$. +Rather than separating budget and non-negativity, gradient steps of the form +\[ + w_{k+1} = \Pi_{\Delta_n}\!\left(w_k - \tau\,\nabla f(w_k)\right) +\] +project directly onto $\Delta_n$ after each descent step, where $\Pi_{\Delta_n}$ +is the Euclidean projection. +Duchi et al.\ \cite{duchi2008} show that $\Pi_{\Delta_n}$ can be computed in +$O(n\log n)$ by a single sort-and-threshold pass, making each iteration cheap. +There is no outer loop: both constraints are handled simultaneously, and the method +terminates when the step size $\|w_{k+1} - w_k\|$ falls below a tolerance. + +The gradient $\nabla f(w) = \tilde{X}^\top(\tilde{X}w) - \tilde{X}^\top\mathbf{0} += \tilde{X}^\top(\tilde{X}w)$ is evaluated matrix-free in $O(nT)$ using two +matrix-vector products with $\tilde{X}$, at the same cost per step as the CG inner +solver. The Lipschitz constant $L = \lambda_{\max}(\tilde{X}^\top\tilde{X})$, needed +for the step size $\tau = 1/(2L)$, is estimated once by power iteration on $\tilde{X}$, +again matrix-free. + +\medskip +The two strategies make different tradeoffs. The active-set loop solves a sequence of +linear systems and benefits from fast inner solvers (Section~\ref{sec:solvers}); its +convergence to global optimality is certified by the KKT check. The simplex-projection +approach is single-stage and loop-free, but converges in $O(\kappa)$ gradient steps +versus $O(\sqrt{\kappa})$ for CG, so it is most competitive when the condition number is +moderate or the matrix-free $O(nT)$ cost dominates. +Section~\ref{sec:solvers} develops concrete implementations of both strategies. \section{Computational Methods}\label{sec:solvers} From ac31b988ddbe26707ec054848269204e70f623c2 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:04:56 +0400 Subject: [PATCH 12/39] refs: add Condat (2016) and cite alongside Duchi for simplex projection Co-Authored-By: Claude Sonnet 4.6 --- paper/bib/refs.bib | 10 ++++++++++ paper/minvar_paper.tex | 13 +++++++------ 2 files changed, 17 insertions(+), 6 deletions(-) diff --git a/paper/bib/refs.bib b/paper/bib/refs.bib index 4681f47..355c7c7 100644 --- a/paper/bib/refs.bib +++ b/paper/bib/refs.bib @@ -181,6 +181,16 @@ @phdthesis{schmelzer2004 type = {Diploma thesis}, } +@article{condat2016, + author = {Condat, Laurent}, + title = {Fast projection onto the simplex and the $\ell_1$ ball}, + journal = {Mathematical Programming}, + year = {2016}, + volume = {158}, + number = {1--2}, + pages = {575--585}, +} + @inproceedings{duchi2008, author = {Duchi, John and Shalev-Shwartz, Shai and Singer, Yoram and Chandra, Tushar}, title = {Efficient Projections onto the $\ell_1$-Ball for Learning in High Dimensions}, diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 55f85c1..075a423 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -271,7 +271,7 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} Section~\ref{sec:pd} reduces the long-only problem to solving an SPD system, leaving $w \geq 0$ as the sole remaining constraint. Two strategies are available. -\paragraph{Active-set strategy.} +\paragraph{Active-set strategy} The first strategy separates the budget constraint from the non-negativity constraint. The budget-only problem is solved over a dynamically adjusted \emph{active set} of assets; non-negativity is then enforced iteratively without ever modifying the inner @@ -320,7 +320,7 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} \end{algorithmic} \end{algorithm} -\paragraph{Simplex-projection strategy.} +\paragraph{Simplex-projection strategy} The second strategy treats the two constraints as a single geometric object: the probability simplex $\Delta_n = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 1,\, w \geq 0\}$. @@ -330,7 +330,7 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} \] project directly onto $\Delta_n$ after each descent step, where $\Pi_{\Delta_n}$ is the Euclidean projection. -Duchi et al.\ \cite{duchi2008} show that $\Pi_{\Delta_n}$ can be computed in +Duchi et al.\ \cite{duchi2008} and Condat~\cite{condat2016} show that $\Pi_{\Delta_n}$ can be computed in $O(n\log n)$ by a single sort-and-threshold pass, making each iteration cheap. There is no outer loop: both constraints are handled simultaneously, and the method terminates when the step size $\|w_{k+1} - w_k\|$ falls below a tolerance. @@ -343,15 +343,16 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} again matrix-free. \medskip -The two strategies make different tradeoffs. The active-set loop solves a sequence of +The two strategies make different tradeoffs. The active-set loop solves a sequence of linear systems and benefits from fast inner solvers (Section~\ref{sec:solvers}); its -convergence to global optimality is certified by the KKT check. The simplex-projection +convergence to global optimality is certified by the KKT check. The simplex-projection approach is single-stage and loop-free, but converges in $O(\kappa)$ gradient steps versus $O(\sqrt{\kappa})$ for CG, so it is most competitive when the condition number is moderate or the matrix-free $O(nT)$ cost dominates. Section~\ref{sec:solvers} develops concrete implementations of both strategies. + \section{Computational Methods}\label{sec:solvers} \subsection{CG Method}\label{ssec:cg} @@ -556,7 +557,7 @@ \subsection{Proximal Gradient}\label{ssec:proximal} where $\Pi_{\Delta_n}$ denotes Euclidean projection onto the simplex. \paragraph{Simplex projection.} -Duchi et al.\ \cite{duchi2008} give an $O(n\log n)$ closed-form algorithm for $\Pi_{\Delta_n}$: +Duchi et al.\ \cite{duchi2008} and Condat~\cite{condat2016} give an $O(n\log n)$ closed-form algorithm for $\Pi_{\Delta_n}$: sort $v$ in decreasing order to obtain $\mu_1 \geq \cdots \geq \mu_n$, find the largest index $\rho$ such that $\mu_\rho > \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, set $\theta = \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, and return From 74a582629ffde52929476215eefe3fccf221c116 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:11:12 +0400 Subject: [PATCH 13/39] paper: move LW shrinkage section before computational methods Readers need to know what X-tilde is before seeing it in the algorithms. Also updates the intro section-range reference and the forward-pointer "solvers above" -> "solvers in Section 6". Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 143 +++++++++++++++++++++-------------------- 1 file changed, 72 insertions(+), 71 deletions(-) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 075a423..e6be784 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -135,7 +135,7 @@ \section{Introduction} Clarabel~\cite{clarabel2024} handle the long-only constraint natively but require as input an estimate of the covariance matrix. The remainder of the paper is organized as follows. -\mbox{Sections~\ref{sec:problem}--\ref{sec:precond}} develop the formulation and solvers; Section~\ref{sec:results} reports +\mbox{Sections~\ref{sec:problem}--\ref{sec:solvers}} develop the formulation, shrinkage, and solvers; Section~\ref{sec:results} reports empirical results on both synthetic and S\&P~500 data. \section{Problem Formulation}\label{sec:problem} @@ -353,6 +353,77 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} +\section{Ledoit-Wolf Shrinkage} +\label{sec:precond} + +It is well known that the sample covariance matrix $\hat \Sigma = \frac{1}{T}X^\top X$ is a poor estimator +of the true covariance matrix $\Sigma$ when $n$ is ``not small" relative to $T$: its smallest +eigenvalues are severely downward-biased, and the condition number (the ratio of largest to smallest eigenvalue) generally far exceeds that of the true covariance matrix. +Ledoit and Wolf \cite{ledoit2004} propose replacing the sample covariance matrix with a convex +combination of itself and the scaled identity matrix (called the {\em shrinkage target}): +\[ +\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha\cdot \bar \lambda I \quad \mbox{with} \quad +\bar\lambda = \frac{\|X\|_{\text{F}}^2}{nT}~. +\] +Here $\|\cdot\|_{\text{F}}$ denotes the Frobenius norm of a square matrix, and +$\bar\lambda$ thus denotes the average eigenvalue of the sample covariance matrix. +The implementation \tcr{Should we say "Our implementation"?} +uses a generalized $N\times N$ regularization shrinkage target matrix $R$ +(satisfying $\mathrm{target} = R^\top R$), \tcr{The target cannot satisfy both target = $R$ and +target = $R^\top R$} +so that +$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha \cdot R^\top R$. +\tcr{Maybe we need a more general notation here, to allow for a generalized shrinkage target.} +Classical Ledoit-Wolf shrinkage is recovered by setting $R = \sqrt{\bar\lambda}\,I$; for i.i.d.\ $\mathcal{N}(0,1)$ +synthetic returns $\bar\lambda \approx 1$ so $R = I$ is equivalent. +The shrinkage intensity $\alpha \in (0,1)$ is +chosen analytically to minimize the expected Frobenius distance to the true covariance matrix, requiring no cross-validation. \tcr{This is not quite true. We are consistently estimating the optimal $\alpha$ only, we cannot guarantee to minimize the expected F-norm distance in finite samples.} +The effect on the spectrum is direct: the target term lifts every eigenvalue, bounding the smallest away from zero and +compressing the overall spread. \tcr{Not sure what is meant by ``lifts every eigenvalue". Classical LW shrinkage moves all sample eigenvalues towards their average. So large eigenvalues go down, whereas small eigenvalues go up.} +The condition number satisfies: +\[ +\kappa(\hat \Sigma_{\text{LW}}) = \frac{\lambda_{\max}(\hat \Sigma_{\text{LW}})}{\lambda_{\min}(\hat \Sigma_{\text{LW}})} \leq +\frac{(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)}{\alpha\lambda_{\min}(R^\top R)}~, +\] +since $\lambda_{\min}(\hat \Sigma_{\text{LW}}) \geq \alpha\lambda_{\min}(R^\top R)$ and, by Weyl's inequality applied to +$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot \hat\Sigma + \alpha \cdot R^\top R$, $\lambda_{\max}(\hat \Sigma_{\text{LW}}) \leq +(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)$, +where $\lambda_{\max}(\hat\Sigma) = \lambda_{\max}(X^\top X/T)$ is +the largest sample eigenvalue. As $\alpha \to 1$, the bound approaches $\lambda_{\max}(R^\top R)/\lambda_{\min}(R^\top R)$, +which equals $1$ for the classical choice $R = \sqrt{\bar\lambda}\,I$, correctly reflecting that full shrinkage +yields a scalar multiple of the identity. For fixed data, the condition number decreases monotonically as $\alpha$ +increases, with the rate governed by the spectral ratio $\lambda_{\max}(\hat\Sigma)/\lambda_{\min}(R^\top R)$. For CG applied to +$\hat \Sigma_{\text{LW}}\,v = \mathbf{1}$, the convergence rate is governed by $\kappa(\hat \Sigma_{\text{LW}})$; compressing the +spectrum directly reduces iteration counts. Subsequent work by the same authors provides an analytical nonlinear +shrinkage scheme that applies a different coefficient to each sample eigenvalue, typically achieving a smaller +Frobenius loss than linear shrinkage when $n/T$ is not negligible; see \cite{ledoit2020}. +Unfortunately, nonlinear shrinkage cannot be expressed as a simple augmented matrix +$\tilde{X}$, so incorporating it into the matrix-free framework would require a separate Krylov-compatible +preconditioner rather than a modified return matrix. + +In the implementation, shrinkage is applied by augmenting the return matrix rather than forming $\hat \Sigma_{\text{LW}}$ +explicitly. Setting $c = 1 - \alpha$, the stacked matrix $\tilde{X} += [\sqrt{c/T}\,X;\,\sqrt{\alpha}\,R]$ satisfies $\tilde{X}^\top \tilde{X} = \frac{c}{T}\,X^\top X + \alpha\,R^\top R += c\,\hat\Sigma + \alpha\,R^\top R = \hat \Sigma_{\text{LW}}$, +so any of the solvers in Section~\ref{sec:solvers} can be applied to $\tilde{X}$ and produce exactly the LW-shrinkage portfolio weights. +For $R = \sqrt{\bar\lambda}\,I$, this results in the classical Ledoit-Wolf linear shrinkage: $\hat \Sigma_{\text{LW}} = (1-\alpha) \cdot \hat\Sigma + \alpha \cdot \bar\lambda\,I$. + +From the perspective of numerical linear algebra, compressing the eigenvalue spectrum of the system matrix is precisely +what a preconditioner does. Ledoit-Wolf shrinkage therefore plays a dual role: it reduces statistical estimation error +and, as a side effect, accelerates iterative solvers by reducing the condition number of the input matrix relative the sample covariance matrix. The two motivations are independent +but arrive at the same modified matrix. \tcr{I do not quite understand the meaning of this last sentence.} + +The proportional identity shrinkage target $\alpha \cdot \bar\lambda I$ is the simplest choice, but the preconditioning +literature suggests sharper alternatives. A factor-model preconditioner, replacing $\alpha \cdot \bar\lambda I$ with a +low-rank-plus-diagonal approximation $BB^\top + D$, would cluster eigenvalues more tightly at the cost of a slightly +more expensive application. Incomplete Cholesky factorization of the shrunk matrix provides a general-purpose option. +For rolling-window problems where the estimated covariance matrix changes slowly between periods, a preconditioner built from the +previous period's factorization could be reused with minimal update cost. None of these +\tcr{none of these what, approaches, methods, \ldots?} +require forming the covariance +matrix explicitly and each is compatible with the matrix-free framework described above. + + \section{Computational Methods}\label{sec:solvers} \subsection{CG Method}\label{ssec:cg} @@ -631,76 +702,6 @@ \subsection{Proximal Gradient}\label{ssec:proximal} implementation at the cost of slower per-problem convergence. } -\section{Ledoit-Wolf Shrinkage} -\label{sec:precond} - -It is well known that the sample covariance matrix $\hat \Sigma = \frac{1}{T}X^\top X$ is a poor estimator -of the true covariance matrix $\Sigma$ when $n$ is ``not small" relative to $T$: its smallest -eigenvalues are severely downward-biased, and the condition number (the ratio of largest to smallest eigenvalue) generally far exceeds that of the true covariance matrix. -Ledoit and Wolf \cite{ledoit2004} propose replacing the sample covariance matrix with a convex -combination of itself and the scaled identity matrix (called the {\em shrinkage target}): -\[ -\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha\cdot \bar \lambda I \quad \mbox{with} \quad -\bar\lambda = \frac{\|X\|_{\text{F}}^2}{nT}~. -\] -Here $\|\cdot\|_{\text{F}}$ denotes the Frobenius norm of a square matrix, and -$\bar\lambda$ thus denotes the average eigenvalue of the sample covariance matrix. -The implementation \tcr{Should we say "Our implementation"?} -uses a generalized $N\times N$ regularization shrinkage target matrix $R$ -(satisfying $\mathrm{target} = R^\top R$), \tcr{The target cannot satisfy both target = $R$ and -target = $R^\top R$} -so that -$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha \cdot R^\top R$. -\tcr{Maybe we need a more general notation here, to allow for a generalized shrinkage target.} -Classical Ledoit-Wolf shrinkage is recovered by setting $R = \sqrt{\bar\lambda}\,I$; for i.i.d.\ $\mathcal{N}(0,1)$ -synthetic returns $\bar\lambda \approx 1$ so $R = I$ is equivalent. -The shrinkage intensity $\alpha \in (0,1)$ is -chosen analytically to minimize the expected Frobenius distance to the true covariance matrix, requiring no cross-validation. \tcr{This is not quite true. We are consistently estimating the optimal $\alpha$ only, we cannot guarantee to minimize the expected F-norm distance in finite samples.} -The effect on the spectrum is direct: the target term lifts every eigenvalue, bounding the smallest away from zero and -compressing the overall spread. \tcr{Not sure what is meant by ``lifts every eigenvalue". Classical LW shrinkage moves all sample eigenvalues towards their average. So large eigenvalues go down, whereas small eigenvalues go up.} -The condition number satisfies: -\[ -\kappa(\hat \Sigma_{\text{LW}}) = \frac{\lambda_{\max}(\hat \Sigma_{\text{LW}})}{\lambda_{\min}(\hat \Sigma_{\text{LW}})} \leq -\frac{(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)}{\alpha\lambda_{\min}(R^\top R)}~, -\] -since $\lambda_{\min}(\hat \Sigma_{\text{LW}}) \geq \alpha\lambda_{\min}(R^\top R)$ and, by Weyl's inequality applied to -$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot \hat\Sigma + \alpha \cdot R^\top R$, $\lambda_{\max}(\hat \Sigma_{\text{LW}}) \leq -(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)$, -where $\lambda_{\max}(\hat\Sigma) = \lambda_{\max}(X^\top X/T)$ is -the largest sample eigenvalue. As $\alpha \to 1$, the bound approaches $\lambda_{\max}(R^\top R)/\lambda_{\min}(R^\top R)$, -which equals $1$ for the classical choice $R = \sqrt{\bar\lambda}\,I$, correctly reflecting that full shrinkage -yields a scalar multiple of the identity. For fixed data, the condition number decreases monotonically as $\alpha$ -increases, with the rate governed by the spectral ratio $\lambda_{\max}(\hat\Sigma)/\lambda_{\min}(R^\top R)$. For CG applied to -$\hat \Sigma_{\text{LW}}\,v = \mathbf{1}$, the convergence rate is governed by $\kappa(\hat \Sigma_{\text{LW}})$; compressing the -spectrum directly reduces iteration counts. Subsequent work by the same authors provides an analytical nonlinear -shrinkage scheme that applies a different coefficient to each sample eigenvalue, typically achieving a smaller -Frobenius loss than linear shrinkage when $n/T$ is not negligible; see \cite{ledoit2020}. -Unfortunately, nonlinear shrinkage cannot be expressed as a simple augmented matrix -$\tilde{X}$, so incorporating it into the matrix-free framework would require a separate Krylov-compatible -preconditioner rather than a modified return matrix. - -In the implementation, shrinkage is applied by augmenting the return matrix rather than forming $\hat \Sigma_{\text{LW}}$ -explicitly. Setting $c = 1 - \alpha$, the stacked matrix $\tilde{X} -= [\sqrt{c/T}\,X;\,\sqrt{\alpha}\,R]$ satisfies $\tilde{X}^\top \tilde{X} = \frac{c}{T}\,X^\top X + \alpha\,R^\top R -= c\,\hat\Sigma + \alpha\,R^\top R = \hat \Sigma_{\text{LW}}$, -so any of the solvers above can be applied to $\tilde{X}$ and produce exactly the LW-shrinkage portfolio weights. -For $R = \sqrt{\bar\lambda}\,I$, this results in the classical Ledoit-Wolf linear shrinkage: $\hat \Sigma_{\text{LW}} = (1-\alpha) \cdot \hat\Sigma + \alpha \cdot \bar\lambda\,I$. - -From the perspective of numerical linear algebra, compressing the eigenvalue spectrum of the system matrix is precisely -what a preconditioner does. Ledoit-Wolf shrinkage therefore plays a dual role: it reduces statistical estimation error -and, as a side effect, accelerates iterative solvers by reducing the condition number of the input matrix relative the sample covariance matrix. The two motivations are independent -but arrive at the same modified matrix. \tcr{I do not quite understand the meaning of this last sentence.} - -The proportional identity shrinkage target $\alpha \cdot \bar\lambda I$ is the simplest choice, but the preconditioning -literature suggests sharper alternatives. A factor-model preconditioner, replacing $\alpha \cdot \bar\lambda I$ with a -low-rank-plus-diagonal approximation $BB^\top + D$, would cluster eigenvalues more tightly at the cost of a slightly -more expensive application. Incomplete Cholesky factorization of the shrunk matrix provides a general-purpose option. -For rolling-window problems where the estimated covariance matrix changes slowly between periods, a preconditioner built from the -previous period's factorization could be reused with minimal update cost. None of these -\tcr{none of these what, approaches, methods, \ldots?} -require forming the covariance -matrix explicitly and each is compatible with the matrix-free framework described above. - \section{Numerical Results}\label{sec:results} All experiments were run on an Apple M4 Pro (14-core CPU, 48\,GB RAM) under Python~3.12, NumPy~2.4, SciPy~1.17, From 35b0e0f9d1246b8a90abd1c7179ba121f02aef0a Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:16:56 +0400 Subject: [PATCH 14/39] paper: fix LaTeX compilation errors - Define \tbr (missing command, brown text for to-be-revised blocks) - Redefine \tcr/\tcb/\tcm/\tcg to use {\color{X}#1} instead of \textcolor{X}{#1} so they can span paragraph breaks - Fix unclosed $ in Algorithm 3 (Direct Solve) algorithmic block Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index e6be784..332d143 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -8,10 +8,11 @@ \usepackage{algpseudocode} \usepackage{authblk} -\newcommand{\tcr}{\textcolor{red}} -\newcommand{\tcm}{\textcolor{magenta}} -\newcommand{\tcg}{\textcolor{green}} -\newcommand{\tcb}{\textcolor{blue}} +\newcommand{\tcr}[1]{{\color{red}#1}} +\newcommand{\tcm}[1]{{\color{magenta}#1}} +\newcommand{\tcg}[1]{{\color{green}#1}} +\newcommand{\tcb}[1]{{\color{blue}#1}} +\newcommand{\tbr}[1]{{\color{brown}#1}} \newcommand{\norm}[1]{\lVert #1 \rVert} @@ -568,7 +569,7 @@ \subsection{Direct Solve (Baseline)} \Function{DirectSolve}{active} \State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $n_a \leftarrow |\mathrm{active}|$ - \State Assemble $\Sigma_{\mathrm{a}} \leftarrow \tfrac{1}{T}\,X_{\mathrm{a}}^\top X_{\mathrm{a}} + \State Assemble $\Sigma_{\mathrm{a}} \leftarrow \tfrac{1}{T}\,X_{\mathrm{a}}^\top X_{\mathrm{a}}$ \hfill\textit{(SPD, $n_a\times n_a$)} \State Solve $\Sigma_{\mathrm{a}}\, v = \mathbf{1}$ by dense factorization \State \Return $v\,/\,(\mathbf{1}^\top v)$,\; $1$ \hfill\textit{(per-step count; no inner iterations)} From 52d06dda44eb50c7a8e620ae633b4ec558310f58 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:20:45 +0400 Subject: [PATCH 15/39] paper: rewrite abstract to include proximal/simplex-projection strategy Abstract now introduces both matrix-free approaches in parallel: CG + active-set (O(sqrt(kappa))) and simplex-projection proximal (loop-free, O(kappa)), with LW shrinkage reducing kappa for both. Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 59 ++++++++++++++++++++---------------------- 1 file changed, 28 insertions(+), 31 deletions(-) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 332d143..d0aee15 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -53,26 +53,31 @@ \begin{abstract} The long-only minimum variance portfolio is routinely solved by assembling a sample covariance -matrix and dispatching it to a general-purpose quadratic program. Both steps are unnecessary -and expensive. - -Conjugate gradients applied matrix-free to the returns matrix $X$ solve the problem in $O(nT)$ -work per iteration without ever forming the $n\times n$ covariance; a primal-dual active-set -loop enforces non-negativity with negligible overhead. The covariance matrix is not a primitive -of the problem---it is a computational detour. +matrix and dispatching it to a general-purpose quadratic program. Both steps are unnecessary: +the covariance matrix is not a primitive of the problem---it is a computational detour. + +We propose two matrix-free strategies that operate directly on the returns matrix $X$ and +never form the $n\times n$ covariance $\hat\Sigma = X^\top X/T$. The first applies conjugate +gradients within a primal-dual active-set loop: each CG step evaluates $\hat\Sigma v$ as two +matrix-vector products with $X$ at $O(nT)$ cost, and the outer loop enforces non-negativity +by iterating over a shrinking active set with a certified KKT stopping criterion. The second +treats the budget and non-negativity constraints as a single object---the probability simplex +$\Delta_n = \{w : \mathbf{1}^\top w = 1,\, w \geq 0\}$---and projects each gradient step +directly onto $\Delta_n$, eliminating the outer loop entirely. The projection costs $O(n\log n)$ +\cite{duchi2008,condat2016} and the gradient remains matrix-free at $O(nT)$. Ledoit-Wolf shrinkage, already standard practice for statistical reasons, turns out to be identical to spectral preconditioning of the linear system: the same ridge term that shrinks -sample eigenvalues toward their mean also compresses the condition number and reduces CG iteration -counts dramatically. Statistical regularization and numerical preconditioning are the same -operation applied to the same matrix, a connection that does not appear to have been made -explicit in the existing literature. +sample eigenvalues toward their mean also compresses the condition number $\kappa$ and reduces +iteration counts for both methods. Statistical regularization and numerical preconditioning +are the same operation applied to the same matrix, a connection that does not appear to have +been made explicit in the existing literature. Numerical experiments on S\&P~500 equity data and synthetic problems of growing size show -order-of-magnitude speedups over interior-point solvers. For standard portfolio sizes a direct -KKT solve already captures most of the gain; the matrix-free advantage of CG becomes decisive -at larger $n$, where avoiding explicit covariance assembly reduces per-step cost from $O(n^2)$ -to $O(Tn)$. +order-of-magnitude speedups over interior-point solvers. For standard portfolio sizes a direct +KKT solve already captures most of the gain; at larger $n$ the matrix-free methods become +decisive, with CG offering $O(\sqrt\kappa)$ convergence and the simplex-projection approach +a loop-free alternative at $O(\kappa)$ convergence. \end{abstract} \section{Introduction} @@ -89,7 +94,6 @@ \section{Introduction} %This paper argues that both steps are unnecessary and expensive. %\tcr{This is a bit confusing. What is meant by the first step? Estimating $\Sigma$? That is certainly necessary. Estimating $\Sigma$ by the sample covariance matrix?} -\tcb{ The Lagrangian for the problem \[ \min_w\; w^\top \hat \Sigma w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, @@ -103,17 +107,8 @@ \section{Introduction} \[ w = v/(\mathbf{1}^\top v), \qquad v = \hat \Sigma^{-1}\mathbf{1}. \] -What has \emph{not} been done is to exploit this structure computationally: the standard pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. -} - -%That the minimum-variance solution satisfies $w \propto \Sigma^{-1}\mathbf{1}$, where $\mathbf{1}$ denotes a conformable vector of ones, -%is a classical observation: it follows -%directly from the KKT stationarity condition $2\Sigma w = \lambda\mathbf{1}$ -%\tcr{At this point, many readers will not know what is meant by KKT.} -%and was made explicit by -%\cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the single rescaling -%$w = v/(\mathbf{1}^\top v)$ where $v = \Sigma^{-1}\mathbf{1}$. What has \emph{not} been done is to exploit this -%structure computationally: the standard pipeline still forms $\Sigma = X^\top X/T$ explicitly before solving. +What has \emph{not} been done is to exploit this structure computationally: the +standard pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. This paper makes two contributions. First, we apply conjugate gradients (CG) to $\hat \Sigma v = \mathbf{1}$ \emph{matrix free}: $\hat \Sigma = X^\top X$ never needs to be assembled explicitly. Each CG iteration requires only two matrix-vector products @@ -128,10 +123,12 @@ \section{Introduction} problem, the same operation applied to the same matrix---a connection that does not appear to have been made explicit in the existing literature. -CG of Hestenes and Stiefel \cite{hestenes1952} is a cornerstone of large-scale scientific computing; e.g., see Golub and van Loan \cite{golub2013}. However, to the best of our knowledge it has not been -applied yet -to portfolio optimisation in the matrix-free form described here. The closest related work applies GMRES to the minimum -variance problem when $\hat \Sigma$ is singular~\cite{bajeux2012}; that formulation still assembles $X^\top X$ and does not +CG of Hestenes and Stiefel \cite{hestenes1952} is a cornerstone of large-scale scientific computing; +e.g., see Golub and van Loan \cite{golub2013}. However, to the best of our knowledge it has not been +applied yet to portfolio optimisation in the matrix-free form described here. +The closest related work applies GMRES to the minimum +variance problem when $\hat \Sigma$ is singular~\cite{bajeux2012}; that +formulation still assembles $X^\top X$ and does not exploit the SPD structure or consider preconditioning. General-purpose solvers such as Clarabel~\cite{clarabel2024} handle the long-only constraint natively but require as input an estimate of the covariance matrix. From accf8583092f0502df4691e0e501fe83cfcf20a3 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:36:07 +0400 Subject: [PATCH 16/39] paper: extract abstract into abstract.tex, \input from main file Co-Authored-By: Claude Sonnet 4.6 --- paper/abstract.tex | 26 ++++++++++++++++++++++++++ paper/minvar_paper.tex | 35 ++--------------------------------- 2 files changed, 28 insertions(+), 33 deletions(-) create mode 100644 paper/abstract.tex diff --git a/paper/abstract.tex b/paper/abstract.tex new file mode 100644 index 0000000..cf5c761 --- /dev/null +++ b/paper/abstract.tex @@ -0,0 +1,26 @@ +The long-only minimum variance portfolio is routinely solved by assembling a sample covariance +matrix and dispatching it to a general-purpose quadratic program. Both steps are unnecessary: +the covariance matrix is not a primitive of the problem. It is a computational detour. + +We propose two matrix-free strategies that operate directly on the returns matrix $X$ and +never form the $n\times n$ covariance $\hat\Sigma = X^\top X/T$. The first applies conjugate +gradients within a primal-dual active-set loop: each CG step evaluates $\hat\Sigma v$ as two +matrix-vector products with $X$ at $O(nT)$ cost, and the outer loop enforces non-negativity +by iterating over a shrinking active set with a certified KKT stopping criterion. The second +treats the budget and non-negativity constraints via the probability simplex +$\Delta_n = \{w : \mathbf{1}^\top w = 1,\, w \geq 0\}$. It projects each gradient step +directly onto $\Delta_n$, eliminating the outer loop entirely. The projection costs $O(n\log n)$ +\cite{duchi2008,condat2016} and the gradient remains matrix-free at $O(nT)$. + +Ledoit-Wolf shrinkage, already standard practice for statistical reasons, turns out to be +identical to spectral preconditioning of the linear system: the same ridge term that shrinks +sample eigenvalues toward their mean also compresses the condition number $\kappa$ and reduces +iteration counts for both methods. Statistical regularization and numerical preconditioning +are the same operation applied to the same matrix, a connection that does not appear to have +been made explicit in the existing literature. + +Numerical experiments on S\&P~500 equity data and synthetic problems of growing size show +order-of-magnitude speedups over interior-point solvers. For standard portfolio sizes a direct +KKT solve already captures most of the gain; at larger $n$ the matrix-free methods become +decisive, with CG offering $O(\sqrt\kappa)$ convergence and the simplex-projection approach +a loop-free alternative at $O(\kappa)$ convergence. diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index d0aee15..79f16e6 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -52,32 +52,7 @@ \maketitle \begin{abstract} -The long-only minimum variance portfolio is routinely solved by assembling a sample covariance -matrix and dispatching it to a general-purpose quadratic program. Both steps are unnecessary: -the covariance matrix is not a primitive of the problem---it is a computational detour. - -We propose two matrix-free strategies that operate directly on the returns matrix $X$ and -never form the $n\times n$ covariance $\hat\Sigma = X^\top X/T$. The first applies conjugate -gradients within a primal-dual active-set loop: each CG step evaluates $\hat\Sigma v$ as two -matrix-vector products with $X$ at $O(nT)$ cost, and the outer loop enforces non-negativity -by iterating over a shrinking active set with a certified KKT stopping criterion. The second -treats the budget and non-negativity constraints as a single object---the probability simplex -$\Delta_n = \{w : \mathbf{1}^\top w = 1,\, w \geq 0\}$---and projects each gradient step -directly onto $\Delta_n$, eliminating the outer loop entirely. The projection costs $O(n\log n)$ -\cite{duchi2008,condat2016} and the gradient remains matrix-free at $O(nT)$. - -Ledoit-Wolf shrinkage, already standard practice for statistical reasons, turns out to be -identical to spectral preconditioning of the linear system: the same ridge term that shrinks -sample eigenvalues toward their mean also compresses the condition number $\kappa$ and reduces -iteration counts for both methods. Statistical regularization and numerical preconditioning -are the same operation applied to the same matrix, a connection that does not appear to have -been made explicit in the existing literature. - -Numerical experiments on S\&P~500 equity data and synthetic problems of growing size show -order-of-magnitude speedups over interior-point solvers. For standard portfolio sizes a direct -KKT solve already captures most of the gain; at larger $n$ the matrix-free methods become -decisive, with CG offering $O(\sqrt\kappa)$ convergence and the simplex-projection approach -a loop-free alternative at $O(\kappa)$ convergence. +\input{abstract} \end{abstract} \section{Introduction} @@ -87,13 +62,6 @@ \section{Introduction} covariance matrix $\Sigma$ of the $n \times 1$ vector of asset returns. In practice, $\Sigma$ is unknown and must be estimated from historical data. To this end, one uses a $T\times n$ matrix of demeaned returns $X$, where $T$ denotes the length of the estimation window (or the sample size). The classical estimator of $\Sigma$ is the sample covariance matrix $\hat \Sigma = X^\top X/T$. -%Since Markowitz \cite{markowitz1952} it has been standard practice %\tcr{Did Markowitz already discuss estimation of $\Sigma$?} \tcb{He only had a $3 \times 3$ matrix in there} to estimate $\Sigma$ by the sample -%covariance matrix $\hat\Sigma = X^\top X / T$ from a $T \times n$ demeaned return matrix $X$ and pass it to a general-purpose -%quadratic programming solver. \tcb{We introduce a method bypassing the explicit computation and storage of the sample covariance matrix $\Sigma$. %It is sufficient to compute the product of $\Sigma v$ which boils down to two matrix-vector products $X^\top X v$. -%} -%This paper argues that both steps are unnecessary and expensive. -%\tcr{This is a bit confusing. What is meant by the first step? Estimating $\Sigma$? That is certainly necessary. Estimating $\Sigma$ by the sample covariance matrix?} - The Lagrangian for the problem \[ \min_w\; w^\top \hat \Sigma w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, @@ -351,6 +319,7 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} + \section{Ledoit-Wolf Shrinkage} \label{sec:precond} From 79f05abe46165c70e6dc2107bb07877c8d1170c5 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:37:27 +0400 Subject: [PATCH 17/39] paper: extract Introduction into s1_introduction.tex Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 49 +-------------------------------------- paper/s1_introduction.tex | 48 ++++++++++++++++++++++++++++++++++++++ 2 files changed, 49 insertions(+), 48 deletions(-) create mode 100644 paper/s1_introduction.tex diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 79f16e6..28149f0 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -55,54 +55,7 @@ \input{abstract} \end{abstract} -\section{Introduction} - -The long-only minimum variance portfolio allocates capital across $n$ assets to minimize portfolio return variance subject to two constraints: the weights must sum to one (the portfolio is fully invested) and must be non-negative. The -only input required in theory is the $n \times n$ -covariance matrix $\Sigma$ of the $n \times 1$ -vector of asset returns. In practice, $\Sigma$ is unknown and must be estimated from historical data. To this end, one uses a $T\times n$ matrix of demeaned returns $X$, where $T$ denotes the length of the estimation window (or the sample size). The classical estimator of $\Sigma$ is the sample covariance matrix $\hat \Sigma = X^\top X/T$. - -The Lagrangian for the problem -\[ -\min_w\; w^\top \hat \Sigma w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, -\] -where $\mathbf{1}$ denotes a conformable vector of ones, is -\[ -\mathcal{L}(w,\lambda) = w^\top \hat \Sigma w - \lambda(\mathbf{1}^\top w - 1), -\] -whose stationarity condition $2 \hat \Sigma w = \lambda\mathbf{1}$ immediately gives $w \propto \hat \Sigma^{-1}\mathbf{1}$; this was made explicit by \cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the -single rescaling -\[ -w = v/(\mathbf{1}^\top v), \qquad v = \hat \Sigma^{-1}\mathbf{1}. -\] -What has \emph{not} been done is to exploit this structure computationally: the -standard pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. - -This paper makes two contributions. First, we apply conjugate gradients (CG) to $\hat \Sigma v = \mathbf{1}$ -\emph{matrix free}: $\hat \Sigma = X^\top X$ never needs to be assembled explicitly. Each CG iteration requires only two matrix-vector products -with $X$, reducing per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. Non-negative weights are -enforced by a primal-dual active-set loop that calls the CG solver on successively smaller subproblems. - -Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004}, a method suggest to come up with a more accurate estimator of the true covariance matrix $\Sigma$ compared to the sample covariance matrix, -plays a dual role in this framework. Statistically, it -reduces estimation error by shrinking sample eigenvalues toward their mean. Numerically, the same modification -compresses the eigenvalue spread of the system matrix and directly accelerates CG convergence. For example, on S\&P~500 equity data, using LW shrinkage instead of the sample covariance matrix reduces the number of CG iterations from 744 to 82. -Statistical regularization and spectral preconditioning are, in this -problem, the same operation applied to the same matrix---a connection that does not appear to have been made explicit in -the existing literature. - -CG of Hestenes and Stiefel \cite{hestenes1952} is a cornerstone of large-scale scientific computing; -e.g., see Golub and van Loan \cite{golub2013}. However, to the best of our knowledge it has not been -applied yet to portfolio optimisation in the matrix-free form described here. -The closest related work applies GMRES to the minimum -variance problem when $\hat \Sigma$ is singular~\cite{bajeux2012}; that -formulation still assembles $X^\top X$ and does not -exploit the SPD structure or consider preconditioning. General-purpose solvers such as -Clarabel~\cite{clarabel2024} handle the long-only constraint natively -but require as input an estimate of the covariance matrix. -The remainder of the paper is organized as follows. -\mbox{Sections~\ref{sec:problem}--\ref{sec:solvers}} develop the formulation, shrinkage, and solvers; Section~\ref{sec:results} reports -empirical results on both synthetic and S\&P~500 data. +\input{s1_introduction} \section{Problem Formulation}\label{sec:problem} The long-only mean-variance portfolio solves diff --git a/paper/s1_introduction.tex b/paper/s1_introduction.tex new file mode 100644 index 0000000..090a8e7 --- /dev/null +++ b/paper/s1_introduction.tex @@ -0,0 +1,48 @@ +\section{Introduction} + +The long-only minimum variance portfolio allocates capital across $n$ assets to minimize portfolio return variance subject to two constraints: the weights must sum to one (the portfolio is fully invested) and must be non-negative. The +only input required in theory is the $n \times n$ +covariance matrix $\Sigma$ of the $n \times 1$ +vector of asset returns. In practice, $\Sigma$ is unknown and must be estimated from historical data. To this end, one uses a $T\times n$ matrix of demeaned returns $X$, where $T$ denotes the length of the estimation window (or the sample size). The classical estimator of $\Sigma$ is the sample covariance matrix $\hat \Sigma = X^\top X/T$. + +The Lagrangian for the problem +\[ +\min_w\; w^\top \hat \Sigma w \quad \text{subject to} \quad \mathbf{1}^\top w = 1, +\] +where $\mathbf{1}$ denotes a conformable vector of ones, is +\[ +\mathcal{L}(w,\lambda) = w^\top \hat \Sigma w - \lambda(\mathbf{1}^\top w - 1), +\] +whose stationarity condition $2 \hat \Sigma w = \lambda\mathbf{1}$ immediately gives $w \propto \hat \Sigma^{-1}\mathbf{1}$; this was made explicit by \cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the +single rescaling +\[ +w = v/(\mathbf{1}^\top v), \qquad v = \hat \Sigma^{-1}\mathbf{1}. +\] +What has \emph{not} been done is to exploit this structure computationally: the +standard pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. + +This paper makes two contributions. First, we apply conjugate gradients (CG) to $\hat \Sigma v = \mathbf{1}$ +\emph{matrix free}: $\hat \Sigma = X^\top X$ never needs to be assembled explicitly. Each CG iteration requires only two matrix-vector products +with $X$, reducing per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. Non-negative weights are +enforced by a primal-dual active-set loop that calls the CG solver on successively smaller subproblems. + +Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004}, a method suggest to come up with a more accurate estimator of the true covariance matrix $\Sigma$ compared to the sample covariance matrix, +plays a dual role in this framework. Statistically, it +reduces estimation error by shrinking sample eigenvalues toward their mean. Numerically, the same modification +compresses the eigenvalue spread of the system matrix and directly accelerates CG convergence. For example, on S\&P~500 equity data, using LW shrinkage instead of the sample covariance matrix reduces the number of CG iterations from 744 to 82. +Statistical regularization and spectral preconditioning are, in this +problem, the same operation applied to the same matrix---a connection that does not appear to have been made explicit in +the existing literature. + +CG of Hestenes and Stiefel \cite{hestenes1952} is a cornerstone of large-scale scientific computing; +e.g., see Golub and van Loan \cite{golub2013}. However, to the best of our knowledge it has not been +applied yet to portfolio optimisation in the matrix-free form described here. +The closest related work applies GMRES to the minimum +variance problem when $\hat \Sigma$ is singular~\cite{bajeux2012}; that +formulation still assembles $X^\top X$ and does not +exploit the SPD structure or consider preconditioning. General-purpose solvers such as +Clarabel~\cite{clarabel2024} handle the long-only constraint natively +but require as input an estimate of the covariance matrix. +The remainder of the paper is organized as follows. +\mbox{Sections~\ref{sec:problem}--\ref{sec:solvers}} develop the formulation, shrinkage, and solvers; Section~\ref{sec:results} reports +empirical results on both synthetic and S\&P~500 data. From 68abde32da6622b08ab5ff0a10f7cf755880c7c5 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:39:56 +0400 Subject: [PATCH 18/39] paper: extract Sections 2-3 into s2s3_problem.tex Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 127 +---------------------------------------- paper/s2s3_problem.tex | 126 ++++++++++++++++++++++++++++++++++++++++ 2 files changed, 127 insertions(+), 126 deletions(-) create mode 100644 paper/s2s3_problem.tex diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 28149f0..9358e7a 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -57,132 +57,7 @@ \input{s1_introduction} -\section{Problem Formulation}\label{sec:problem} -The long-only mean-variance portfolio solves -\[ -\min_{w}\; w^\top\Sigma\, w - \rho\,\mu^\top w -\quad\text{subject to}\quad -\mathbf{1}^\top w = 1,\quad w\geq 0~, -\] -where $\Sigma$ is the $n\times n$ covariance matrix of returns, $\mu$ is the $n\times 1$ vector of expected returns, and $\rho\geq 0$ controls the risk-return trade-off; setting $\rho=0$ recovers the pure long-only minimum variance portfolio. -Since both $\Sigma$ and -$\mu$ are unknown, in practice they must be replaced by estimates. Let $X\in\mathbb{R}^{T\times n}$ denote the demeaned return matrix for $n$ assets over the previous $T$ trading periods. The feasible problem formulation then is -\[ -\min_{w}\; w^\top\hat\Sigma\, w - \rho\,\hat\mu^\top w -\quad\text{subject to}\quad -\mathbf{1}^\top w = 1,\quad w\geq 0~. -\] -The classical choice for estimating $\Sigma$ -the sample covariance matrix $\hat\Sigma = X^\top X/T$, for which $w^\top\hat\Sigma\, w = \norm{Xw}^2/T$, and the problem becomes -\[ -\min_{w}\;\frac{\norm{Xw}^2}{T} - \rho\,\hat\mu^\top w -\quad\text{subject to}\quad -\mathbf{1}^\top w = 1,\quad w\geq 0~. -\] -This reframing is key: the variance term is now a normalised squared norm of $Xw$, and the gradient $w\mapsto X^\top(Xw)$ can be evaluated with two matrix--vector products, without ever forming $X^\top X$. - -\begin{table}[ht] -\centering -\begin{tabular}{lll} -\toprule -Parameter & Type & Description \\ -\midrule -$X$ & $\mathbb{R}^{T \times n}$ & Demeaned return matrix \\ -$\rho$ & $\mathbb{R}_{\geq 0}$ & Return-tilt weight (default $0$) \\ -$\mu$ & $\mathbb{R}^n$ & Expected returns (default $\mathbf{0}$) \\ -\bottomrule -\end{tabular} -\caption{Parameters of the \texttt{Problem} specification.} -\label{tab:params} -\end{table} - -\paragraph{Dual problem} -Including multipliers for both constraints, the full Lagrangian is -\[ -\mathcal{L}(w,\, \lambda,\, \nu) -= \frac{\lVert Xw\rVert^2}{T} - \rho\hat\mu^\top w - - \lambda(\mathbf{1}^\top w - 1) - \nu^\top w\;, -\] -where $\lambda\in\mathbb{R}$ is the budget multiplier and $\nu\geq\mathbf{0}$ is the vector of -non-negativity multipliers. The stationarity condition -\[ -\tfrac{2}{T}\, X^\top X w = \lambda\mathbf{1}+\nu+\rho\hat\mu -\] -gives $w^* = \tfrac{T}{2}(X^\top X)^{-1}s$ with $s = \lambda\mathbf{1}+\nu+\rho\hat\mu$. -Substituting into $\mathcal{L}$: -\[ -\mathcal{L}(w^*,\,\lambda,\,\nu) -= \tfrac{1}{T}(w^*)^\top X^\top X w^* - s^\top w^* + \lambda\;. -\] -From stationarity $\tfrac{1}{T}X^\top X w^* = \tfrac{s}{2}$, so the first term equals -$\tfrac{1}{2}s^\top w^*$, giving -\[ -g(\lambda,\,\nu) -= \tfrac{1}{2}s^\top w^* - s^\top w^* + \lambda -= -\tfrac{1}{2}s^\top w^* + \lambda\;. -\] -Substituting $w^* = \tfrac{T}{2}(X^\top X)^{-1}s$ yields the dual function -\[ -g(\lambda,\,\nu) -= \inf_w\; \mathcal{L}(w,\,\lambda,\,\nu) -= \lambda - \tfrac{T}{4}\, - (\lambda\mathbf{1}+\nu+\rho\hat\mu)^\top(X^\top X)^{-1} - (\lambda\mathbf{1}+\nu+\rho\hat\mu)\;, -\] -defined when $X$ has full column rank; the dual problem is -$\max_{\lambda,\,\nu\geq\mathbf{0}}\; g(\lambda,\,\nu)$. -The feasible set $\{w:\mathbf{1}^\top w=1,\,w\geq\mathbf{0}\}$ is compact and -$w=\tfrac{1}{n}\mathbf{1}$ is strictly feasible, so Slater's condition holds and -strong duality obtains. - -At the optimum, complementary slackness ($\nu_i w_i = 0$ for all $i$) partitions assets -into two types. \emph{Active} assets ($w_i>0$) have $\nu_i=0$; stationarity then gives -\[ -\tfrac{2}{T}\,(X^\top Xw)_i = \lambda+\rho\hat\mu_i\;, -\] -equating each active asset's marginal variance to the budget shadow price adjusted for its -expected return. \emph{Excluded} assets ($w_i=0$) satisfy -\[ -\nu_i = \tfrac{2}{T}\,(X^\top Xw)_i - \lambda - \rho\hat\mu_i \geq 0~. -\] -Their marginal variance exceeds this threshold, so holding them at a positive weight would -increase variance net of return. The primal--dual loop in Section~\ref{sec:primaldual} is a direct -implementation of these conditions: it terminates precisely when primal and dual feasibility -both hold, which---together with stationarity from each inner solve---is sufficient for -global optimality. - -\section{Positive-Definite Reduction}\label{sec:pd} - -Ignoring the inequality constraints for now, the Lagrangian for the equality-constrained problem is: -\[ -\mathcal{L}(w, \lambda) = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1)~ + \rho \hat \mu~. -\] -Setting the derivative to zero gives the stationarity condition -$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\hat \mu$, where $\hat\Sigma = X^\top X/T$ (or $\hat \Sigma_{\mathrm{LW}}$ with shrinkage). -For $\rho = 0$, this reduces to $\frac{2}{T}\hat\Sigma\, w = \lambda\mathbf{1}$, determining the solution up to a scalar. - -\paragraph{Positive-definite reduction} -When $\rho = 0$ (pure long-only minimum variance), stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1}$, -so $w$ is a scalar multiple of $\hat\Sigma^{-1}\mathbf{1}$. -Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and writing $w = c\,v_1$, the budget constraint -$\mathbf{1}^\top w = 1$ \tcb{scales} -\[ - c = \frac{1}{\mathbf{1}^\top v_1} \quad \mbox{and} \quad w = \frac{v_1}{\mathbf{1}^\top v_1}~. -\] -When $\rho \neq 0$, stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1} + \rho\hat \mu$, so -$w = \tfrac{\lambda}{2}\,\hat\Sigma^{-1}\mathbf{1} + \tfrac{\rho}{2}\,\hat\Sigma^{-1}\hat \mu$. -Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and $v_2 = \hat\Sigma^{-1}\hat \mu$ (two SPD solves), the budget -constraint determines the multiplier -\[ - \lambda = \frac{2 - \rho\,(\mathbf{1}^\top v_2)}{\mathbf{1}^\top v_1}~, -\] -and the weights follow as $w = \tfrac{\lambda}{2}\,v_1 + \tfrac{\rho}{2}\,v_2$. -In both cases the $(n+1)\times(n+1)$ saddle-point system is never needed and $\lambda$ is -recovered analytically rather than as part of the solve. -The long-only problem therefore reduces to solving a symmetric positive definite system -$\hat\Sigma_{\mathrm{LW}}\,v = b$ for one or two right-hand sides, subject to the remaining -inequality constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}, -which presents two strategies for doing so. +\input{s2s3_problem} \section{Enforcing Non-Negativity}\label{sec:primaldual} diff --git a/paper/s2s3_problem.tex b/paper/s2s3_problem.tex new file mode 100644 index 0000000..8eb8b2c --- /dev/null +++ b/paper/s2s3_problem.tex @@ -0,0 +1,126 @@ +\section{Problem Formulation}\label{sec:problem} +The long-only mean-variance portfolio solves +\[ +\min_{w}\; w^\top\Sigma\, w - \rho\,\mu^\top w +\quad\text{subject to}\quad +\mathbf{1}^\top w = 1,\quad w\geq 0~, +\] +where $\Sigma$ is the $n\times n$ covariance matrix of returns, $\mu$ is the $n\times 1$ vector of expected returns, and $\rho\geq 0$ controls the risk-return trade-off; setting $\rho=0$ recovers the pure long-only minimum variance portfolio. +Since both $\Sigma$ and +$\mu$ are unknown, in practice they must be replaced by estimates. Let $X\in\mathbb{R}^{T\times n}$ denote the demeaned return matrix for $n$ assets over the previous $T$ trading periods. The feasible problem formulation then is +\[ +\min_{w}\; w^\top\hat\Sigma\, w - \rho\,\hat\mu^\top w +\quad\text{subject to}\quad +\mathbf{1}^\top w = 1,\quad w\geq 0~. +\] +The classical choice for estimating $\Sigma$ +the sample covariance matrix $\hat\Sigma = X^\top X/T$, for which $w^\top\hat\Sigma\, w = \norm{Xw}^2/T$, and the problem becomes +\[ +\min_{w}\;\frac{\norm{Xw}^2}{T} - \rho\,\hat\mu^\top w +\quad\text{subject to}\quad +\mathbf{1}^\top w = 1,\quad w\geq 0~. +\] +This reframing is key: the variance term is now a normalised squared norm of $Xw$, and the gradient $w\mapsto X^\top(Xw)$ can be evaluated with two matrix--vector products, without ever forming $X^\top X$. + +\begin{table}[ht] +\centering +\begin{tabular}{lll} +\toprule +Parameter & Type & Description \\ +\midrule +$X$ & $\mathbb{R}^{T \times n}$ & Demeaned return matrix \\ +$\rho$ & $\mathbb{R}_{\geq 0}$ & Return-tilt weight (default $0$) \\ +$\mu$ & $\mathbb{R}^n$ & Expected returns (default $\mathbf{0}$) \\ +\bottomrule +\end{tabular} +\caption{Parameters of the \texttt{Problem} specification.} +\label{tab:params} +\end{table} + +\paragraph{Dual problem} +Including multipliers for both constraints, the full Lagrangian is +\[ +\mathcal{L}(w,\, \lambda,\, \nu) += \frac{\lVert Xw\rVert^2}{T} - \rho\hat\mu^\top w + - \lambda(\mathbf{1}^\top w - 1) - \nu^\top w\;, +\] +where $\lambda\in\mathbb{R}$ is the budget multiplier and $\nu\geq\mathbf{0}$ is the vector of +non-negativity multipliers. The stationarity condition +\[ +\tfrac{2}{T}\, X^\top X w = \lambda\mathbf{1}+\nu+\rho\hat\mu +\] +gives $w^* = \tfrac{T}{2}(X^\top X)^{-1}s$ with $s = \lambda\mathbf{1}+\nu+\rho\hat\mu$. +Substituting into $\mathcal{L}$: +\[ +\mathcal{L}(w^*,\,\lambda,\,\nu) += \tfrac{1}{T}(w^*)^\top X^\top X w^* - s^\top w^* + \lambda\;. +\] +From stationarity $\tfrac{1}{T}X^\top X w^* = \tfrac{s}{2}$, so the first term equals +$\tfrac{1}{2}s^\top w^*$, giving +\[ +g(\lambda,\,\nu) += \tfrac{1}{2}s^\top w^* - s^\top w^* + \lambda += -\tfrac{1}{2}s^\top w^* + \lambda\;. +\] +Substituting $w^* = \tfrac{T}{2}(X^\top X)^{-1}s$ yields the dual function +\[ +g(\lambda,\,\nu) += \inf_w\; \mathcal{L}(w,\,\lambda,\,\nu) += \lambda - \tfrac{T}{4}\, + (\lambda\mathbf{1}+\nu+\rho\hat\mu)^\top(X^\top X)^{-1} + (\lambda\mathbf{1}+\nu+\rho\hat\mu)\;, +\] +defined when $X$ has full column rank; the dual problem is +$\max_{\lambda,\,\nu\geq\mathbf{0}}\; g(\lambda,\,\nu)$. +The feasible set $\{w:\mathbf{1}^\top w=1,\,w\geq\mathbf{0}\}$ is compact and +$w=\tfrac{1}{n}\mathbf{1}$ is strictly feasible, so Slater's condition holds and +strong duality obtains. + +At the optimum, complementary slackness ($\nu_i w_i = 0$ for all $i$) partitions assets +into two types. \emph{Active} assets ($w_i>0$) have $\nu_i=0$; stationarity then gives +\[ +\tfrac{2}{T}\,(X^\top Xw)_i = \lambda+\rho\hat\mu_i\;, +\] +equating each active asset's marginal variance to the budget shadow price adjusted for its +expected return. \emph{Excluded} assets ($w_i=0$) satisfy +\[ +\nu_i = \tfrac{2}{T}\,(X^\top Xw)_i - \lambda - \rho\hat\mu_i \geq 0~. +\] +Their marginal variance exceeds this threshold, so holding them at a positive weight would +increase variance net of return. The primal--dual loop in Section~\ref{sec:primaldual} is a direct +implementation of these conditions: it terminates precisely when primal and dual feasibility +both hold, which---together with stationarity from each inner solve---is sufficient for +global optimality. + +\section{Positive-Definite Reduction}\label{sec:pd} + +Ignoring the inequality constraints for now, the Lagrangian for the equality-constrained problem is: +\[ +\mathcal{L}(w, \lambda) = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1)~ + \rho \hat \mu~. +\] +Setting the derivative to zero gives the stationarity condition +$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\hat \mu$, where $\hat\Sigma = X^\top X/T$ (or $\hat \Sigma_{\mathrm{LW}}$ with shrinkage). +For $\rho = 0$, this reduces to $\frac{2}{T}\hat\Sigma\, w = \lambda\mathbf{1}$, determining the solution up to a scalar. + +\paragraph{Positive-definite reduction} +When $\rho = 0$ (pure long-only minimum variance), stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1}$, +so $w$ is a scalar multiple of $\hat\Sigma^{-1}\mathbf{1}$. +Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and writing $w = c\,v_1$, the budget constraint +$\mathbf{1}^\top w = 1$ \tcb{scales} +\[ + c = \frac{1}{\mathbf{1}^\top v_1} \quad \mbox{and} \quad w = \frac{v_1}{\mathbf{1}^\top v_1}~. +\] +When $\rho \neq 0$, stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1} + \rho\hat \mu$, so +$w = \tfrac{\lambda}{2}\,\hat\Sigma^{-1}\mathbf{1} + \tfrac{\rho}{2}\,\hat\Sigma^{-1}\hat \mu$. +Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and $v_2 = \hat\Sigma^{-1}\hat \mu$ (two SPD solves), the budget +constraint determines the multiplier +\[ + \lambda = \frac{2 - \rho\,(\mathbf{1}^\top v_2)}{\mathbf{1}^\top v_1}~, +\] +and the weights follow as $w = \tfrac{\lambda}{2}\,v_1 + \tfrac{\rho}{2}\,v_2$. +In both cases the $(n+1)\times(n+1)$ saddle-point system is never needed and $\lambda$ is +recovered analytically rather than as part of the solve. +The long-only problem therefore reduces to solving a symmetric positive definite system +$\hat\Sigma_{\mathrm{LW}}\,v = b$ for one or two right-hand sides, subject to the remaining +inequality constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}, +which presents two strategies for doing so. From d6f4cf02a14d125f1709c7badf9110ebcee224b4 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:41:45 +0400 Subject: [PATCH 19/39] paper: add Algorithm 2 (proximal gradient) in Section 4 Gives the simplex-projection strategy its own algorithm alongside Algorithm 1 (active-set), with matrix-free gradient and O(n log n) simplex projection explicitly annotated per step. Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 9358e7a..0403c2d 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -136,6 +136,29 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} for the step size $\tau = 1/(2L)$, is estimated once by power iteration on $\tilde{X}$, again matrix-free. +\begin{algorithm}[ht] +\caption{Simplex-projection solver (proximal gradient, matrix-free)}\label{alg:proximal} +\begin{algorithmic}[1] +\Require Stacked matrix $\tilde{X} \in \mathbb{R}^{(T+n_R)\times n}$;\; + tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ +\Ensure Weight vector $w \in \Delta_n$ +\State Estimate $L \leftarrow \lambda_{\max}(\tilde{X}^\top\tilde{X})$ by power iteration + \hfill\textit{(matrix-free, $O(n_{\mathrm{pow}}\cdot nT)$)} +\State $\tau \leftarrow (2L)^{-1}$;\; initialise $w_0 \in \mathbb{R}^n$ randomly;\; $k \leftarrow 0$ +\Loop + \State $g \leftarrow \tilde{X}^\top(\tilde{X}w_k)$ + \hfill\textit{(matrix-free gradient, $O(nT)$)} + \State $w_{k+1} \leftarrow \Pi_{\Delta_n}(w_k - \tau\,g)$ + \hfill\textit{(simplex projection, $O(n\log n)$)} + \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$ \textbf{or} $k \geq k_{\max}$} + \textbf{ break} + \EndIf + \State $k \leftarrow k + 1$ +\EndLoop +\State \Return $w_{k+1}$ +\end{algorithmic} +\end{algorithm} + \medskip The two strategies make different tradeoffs. The active-set loop solves a sequence of linear systems and benefits from fast inner solvers (Section~\ref{sec:solvers}); its From 32316f907e47b117237ebb6194bf741bd3cf5d18 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:48:19 +0400 Subject: [PATCH 20/39] paper: extract Section 4 into s4_nonneg.tex Also renames the Section 6.4 proximal algorithm label to alg:proximal_detail to avoid conflict with the new Algorithm 2 (alg:proximal) in s4_nonneg.tex; restores \tbr dropped by linter. Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 112 ++--------------------------------------- paper/s4_nonneg.tex | 107 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 110 insertions(+), 109 deletions(-) create mode 100644 paper/s4_nonneg.tex diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 0403c2d..d43bec0 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -60,113 +60,7 @@ \input{s2s3_problem} -\section{Enforcing Non-Negativity}\label{sec:primaldual} - -Section~\ref{sec:pd} reduces the long-only problem to solving an SPD system, leaving -$w \geq 0$ as the sole remaining constraint. Two strategies are available. - -\paragraph{Active-set strategy} -The first strategy separates the budget constraint from the non-negativity constraint. -The budget-only problem is solved over a dynamically adjusted \emph{active set} of -assets; non-negativity is then enforced iteratively without ever modifying the inner -solve. - -In the \emph{primal step}, any asset with a negative weight is dropped from the active -set and the budget-constrained SPD system is re-solved on the reduced universe. Once -all active weights are non-negative, the \emph{dual step} checks the KKT gradient -condition for every excluded asset: global optimality requires every excluded asset $i$ -to satisfy -\[ - \nabla_i f(w) \;\geq\; \lambda\;, - \qquad - \nabla_i f(w) = \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i\;, -\] -where $\lambda$ is the budget Lagrange multiplier recovered analytically from -Section~\ref{sec:pd}. An excluded asset that violates this condition would decrease -portfolio variance if held at a small positive weight; it is re-added to the active set -and the loop repeats. The loop terminates when both primal and dual feasibility hold; -together with stationarity from the inner solve this certifies global optimality. - -\begin{algorithm}[ht] -\caption{Active-set outer loop (wraps any inner solver $f$)}\label{alg:elim} -\begin{algorithmic}[1] -\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$ -\Ensure Weight vector $w \in \mathbb{R}^n$;\; total inner iteration count $k$ -\State $\mathrm{active} \leftarrow \{1,\ldots,n\}$;\; $k \leftarrow 0$ -\Loop - \State $(w_{\mathrm{a}},\, k_{\mathrm{step}}) \leftarrow f(\mathrm{active})$;\; $k \leftarrow k + k_{\mathrm{step}}$ - \If{$\exists\, i \in \mathrm{active}$ with $w_{a,i} < -\varepsilon$} - \State $\mathrm{active} \leftarrow \mathrm{active} \setminus \{i : w_{a,i} < -\varepsilon\}$ - \hfill\textit{(primal step: drop violators)} - \Else - \State $w_i \leftarrow w_{a,i}$ for $i \in \mathrm{active}$;\; $w_i \leftarrow 0$ for $i \notin \mathrm{active}$ - \State $g_i \leftarrow \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i$ - \State $\lambda \leftarrow \mathrm{mean}_{j \in \mathrm{active}}(g_j)$ - \State $\mathcal{V} \leftarrow \{i \notin \mathrm{active} : g_i < \lambda - \varepsilon\}$ - \If{$\mathcal{V} = \emptyset$} - \State \textbf{break} \hfill\textit{(KKT satisfied; globally optimal)} - \EndIf - \State $\mathrm{active} \leftarrow \mathrm{active} \cup \mathcal{V}$ - \hfill\textit{(dual step: re-add violators)} - \EndIf -\EndLoop -\State \Return $w$, $k$ -\end{algorithmic} -\end{algorithm} - -\paragraph{Simplex-projection strategy} -The second strategy treats the two constraints as a single geometric object: the -probability simplex -$\Delta_n = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 1,\, w \geq 0\}$. -Rather than separating budget and non-negativity, gradient steps of the form -\[ - w_{k+1} = \Pi_{\Delta_n}\!\left(w_k - \tau\,\nabla f(w_k)\right) -\] -project directly onto $\Delta_n$ after each descent step, where $\Pi_{\Delta_n}$ -is the Euclidean projection. -Duchi et al.\ \cite{duchi2008} and Condat~\cite{condat2016} show that $\Pi_{\Delta_n}$ can be computed in -$O(n\log n)$ by a single sort-and-threshold pass, making each iteration cheap. -There is no outer loop: both constraints are handled simultaneously, and the method -terminates when the step size $\|w_{k+1} - w_k\|$ falls below a tolerance. - -The gradient $\nabla f(w) = \tilde{X}^\top(\tilde{X}w) - \tilde{X}^\top\mathbf{0} -= \tilde{X}^\top(\tilde{X}w)$ is evaluated matrix-free in $O(nT)$ using two -matrix-vector products with $\tilde{X}$, at the same cost per step as the CG inner -solver. The Lipschitz constant $L = \lambda_{\max}(\tilde{X}^\top\tilde{X})$, needed -for the step size $\tau = 1/(2L)$, is estimated once by power iteration on $\tilde{X}$, -again matrix-free. - -\begin{algorithm}[ht] -\caption{Simplex-projection solver (proximal gradient, matrix-free)}\label{alg:proximal} -\begin{algorithmic}[1] -\Require Stacked matrix $\tilde{X} \in \mathbb{R}^{(T+n_R)\times n}$;\; - tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ -\Ensure Weight vector $w \in \Delta_n$ -\State Estimate $L \leftarrow \lambda_{\max}(\tilde{X}^\top\tilde{X})$ by power iteration - \hfill\textit{(matrix-free, $O(n_{\mathrm{pow}}\cdot nT)$)} -\State $\tau \leftarrow (2L)^{-1}$;\; initialise $w_0 \in \mathbb{R}^n$ randomly;\; $k \leftarrow 0$ -\Loop - \State $g \leftarrow \tilde{X}^\top(\tilde{X}w_k)$ - \hfill\textit{(matrix-free gradient, $O(nT)$)} - \State $w_{k+1} \leftarrow \Pi_{\Delta_n}(w_k - \tau\,g)$ - \hfill\textit{(simplex projection, $O(n\log n)$)} - \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$ \textbf{or} $k \geq k_{\max}$} - \textbf{ break} - \EndIf - \State $k \leftarrow k + 1$ -\EndLoop -\State \Return $w_{k+1}$ -\end{algorithmic} -\end{algorithm} - -\medskip -The two strategies make different tradeoffs. The active-set loop solves a sequence of -linear systems and benefits from fast inner solvers (Section~\ref{sec:solvers}); its -convergence to global optimality is certified by the KKT check. The simplex-projection -approach is single-stage and loop-free, but converges in $O(\kappa)$ gradient steps -versus $O(\sqrt{\kappa})$ for CG, so it is most competitive when the condition number is -moderate or the matrix-free $O(nT)$ cost dominates. -Section~\ref{sec:solvers} develops concrete implementations of both strategies. +\input{s4_nonneg} @@ -451,10 +345,10 @@ \subsection{Proximal Gradient}\label{ssec:proximal} $\rho$ such that $\mu_\rho > \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, set $\theta = \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, and return $[\Pi_{\Delta_n}(v)]_i = \max(v_i - \theta,\,0)$. The projection requires a single sort and a -linear scan; it is the inner loop of Algorithm~\ref{alg:proximal}. +linear scan; it is the inner loop of Algorithm~\ref{alg:proximal_detail}. \begin{algorithm}[ht] -\caption{Proximal gradient solver}\label{alg:proximal} +\caption{Proximal gradient solver (detailed)}\label{alg:proximal_detail} \begin{algorithmic}[1] \Require Stacked return matrix $\tilde{X} \in \mathbb{R}^{(T+n_R) \times n}$;\; tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ diff --git a/paper/s4_nonneg.tex b/paper/s4_nonneg.tex new file mode 100644 index 0000000..ae770c5 --- /dev/null +++ b/paper/s4_nonneg.tex @@ -0,0 +1,107 @@ +\section{Enforcing Non-Negativity}\label{sec:primaldual} + +Section~\ref{sec:pd} reduces the long-only problem to solving an SPD system, leaving +$w \geq 0$ as the sole remaining constraint. Two strategies are available. + +\paragraph{Active-set strategy} +The first strategy separates the budget constraint from the non-negativity constraint. +The budget-only problem is solved over a dynamically adjusted \emph{active set} of +assets; non-negativity is then enforced iteratively without ever modifying the inner +solve. + +In the \emph{primal step}, any asset with a negative weight is dropped from the active +set and the budget-constrained SPD system is re-solved on the reduced universe. Once +all active weights are non-negative, the \emph{dual step} checks the KKT gradient +condition for every excluded asset: global optimality requires every excluded asset $i$ +to satisfy +\[ + \nabla_i f(w) \;\geq\; \lambda\;, + \qquad + \nabla_i f(w) = \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i\;, +\] +where $\lambda$ is the budget Lagrange multiplier recovered analytically from +Section~\ref{sec:pd}. An excluded asset that violates this condition would decrease +portfolio variance if held at a small positive weight; it is re-added to the active set +and the loop repeats. The loop terminates when both primal and dual feasibility hold; +together with stationarity from the inner solve this certifies global optimality. + +\begin{algorithm}[ht] +\caption{Active-set outer loop (wraps any inner solver $f$)}\label{alg:elim} +\begin{algorithmic}[1] +\Require Inner solver $f$;\; tolerance $\varepsilon = 10^{-6}$ +\Ensure Weight vector $w \in \mathbb{R}^n$;\; total inner iteration count $k$ +\State $\mathrm{active} \leftarrow \{1,\ldots,n\}$;\; $k \leftarrow 0$ +\Loop + \State $(w_{\mathrm{a}},\, k_{\mathrm{step}}) \leftarrow f(\mathrm{active})$;\; $k \leftarrow k + k_{\mathrm{step}}$ + \If{$\exists\, i \in \mathrm{active}$ with $w_{a,i} < -\varepsilon$} + \State $\mathrm{active} \leftarrow \mathrm{active} \setminus \{i : w_{a,i} < -\varepsilon\}$ + \hfill\textit{(primal step: drop violators)} + \Else + \State $w_i \leftarrow w_{a,i}$ for $i \in \mathrm{active}$;\; $w_i \leftarrow 0$ for $i \notin \mathrm{active}$ + \State $g_i \leftarrow \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i$ + \State $\lambda \leftarrow \mathrm{mean}_{j \in \mathrm{active}}(g_j)$ + \State $\mathcal{V} \leftarrow \{i \notin \mathrm{active} : g_i < \lambda - \varepsilon\}$ + \If{$\mathcal{V} = \emptyset$} + \State \textbf{break} \hfill\textit{(KKT satisfied; globally optimal)} + \EndIf + \State $\mathrm{active} \leftarrow \mathrm{active} \cup \mathcal{V}$ + \hfill\textit{(dual step: re-add violators)} + \EndIf +\EndLoop +\State \Return $w$, $k$ +\end{algorithmic} +\end{algorithm} + +\paragraph{Simplex-projection strategy} +The second strategy treats the two constraints as a single geometric object: the +probability simplex +$\Delta_n = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 1,\, w \geq 0\}$. +Rather than separating budget and non-negativity, gradient steps of the form +\[ + w_{k+1} = \Pi_{\Delta_n}\!\left(w_k - \tau\,\nabla f(w_k)\right) +\] +project directly onto $\Delta_n$ after each descent step, where $\Pi_{\Delta_n}$ +is the Euclidean projection. +Duchi et al.\ \cite{duchi2008} and Condat~\cite{condat2016} show that $\Pi_{\Delta_n}$ can be computed in +$O(n\log n)$ by a single sort-and-threshold pass, making each iteration cheap. +There is no outer loop: both constraints are handled simultaneously, and the method +terminates when the step size $\|w_{k+1} - w_k\|$ falls below a tolerance. + +The gradient $\nabla f(w) = \tilde{X}^\top(\tilde{X}w) - \tilde{X}^\top\mathbf{0} += \tilde{X}^\top(\tilde{X}w)$ is evaluated matrix-free in $O(nT)$ using two +matrix-vector products with $\tilde{X}$, at the same cost per step as the CG inner +solver. The Lipschitz constant $L = \lambda_{\max}(\tilde{X}^\top\tilde{X})$, needed +for the step size $\tau = 1/(2L)$, is estimated once by power iteration on $\tilde{X}$, +again matrix-free. + +\begin{algorithm}[ht] +\caption{Simplex-projection solver (proximal gradient, matrix-free)}\label{alg:proximal} +\begin{algorithmic}[1] +\Require Stacked matrix $\tilde{X} \in \mathbb{R}^{(T+n_R)\times n}$;\; + tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ +\Ensure Weight vector $w \in \Delta_n$ +\State Estimate $L \leftarrow \lambda_{\max}(\tilde{X}^\top\tilde{X})$ by power iteration + \hfill\textit{(matrix-free, $O(n_{\mathrm{pow}}\cdot nT)$)} +\State $\tau \leftarrow (2L)^{-1}$;\; initialise $w_0 \in \mathbb{R}^n$ randomly;\; $k \leftarrow 0$ +\Loop + \State $g \leftarrow \tilde{X}^\top(\tilde{X}w_k)$ + \hfill\textit{(matrix-free gradient, $O(nT)$)} + \State $w_{k+1} \leftarrow \Pi_{\Delta_n}(w_k - \tau\,g)$ + \hfill\textit{(simplex projection, $O(n\log n)$)} + \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$ \textbf{or} $k \geq k_{\max}$} + \textbf{ break} + \EndIf + \State $k \leftarrow k + 1$ +\EndLoop +\State \Return $w_{k+1}$ +\end{algorithmic} +\end{algorithm} + +\medskip +The two strategies make different tradeoffs. The active-set loop solves a sequence of +linear systems and benefits from fast inner solvers (Section~\ref{sec:solvers}); its +convergence to global optimality is certified by the KKT check. The simplex-projection +approach is single-stage and loop-free, but converges in $O(\kappa)$ gradient steps +versus $O(\sqrt{\kappa})$ for CG, so it is most competitive when the condition number is +moderate or the matrix-free $O(nT)$ cost dominates. +Section~\ref{sec:solvers} develops concrete implementations of both strategies. From 1852f82b819e3dc15118a49aa9e7dfc61ed5f94c Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 19:50:07 +0400 Subject: [PATCH 21/39] paper: collapse NNLS subsection to a single paragraph Removes the subsection heading, algorithm box, and editorial \tcb{} wrapper. Retains the key observation (paradoxically slower with shrinkage) as a \paragraph{NNLS baseline.} before the Proximal Gradient subsection. Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 33 ++++++++++----------------------- 1 file changed, 10 insertions(+), 23 deletions(-) diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index d43bec0..85feade 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -293,29 +293,16 @@ \subsection{Direct Solve (Baseline)} -\subsection{NNLS Solver}\label{ssec:nnls} -\tcb{But NNLS is using an active-set method under the hood, too -An alternative approach bypasses the primal-dual loop entirely by reformulating the long-only constraint as a -non-negative least squares (NNLS) problem. Starting from the objective $\min_w \frac{1-\alpha}{T}\|Xw\|^2 + -\alpha\|Rw\|^2$, the budget constraint $\mathbf{1}^\top w = 1$ is incorporated via a large-penalty row, yielding the -augmented system: -\[ - \min_{w \geq 0}\; \left\|\begin{pmatrix} \sqrt{1/T}\,X \\ M\,\mathbf{1}^\top \end{pmatrix} - w - \begin{pmatrix} \mathbf{0} \\ M \end{pmatrix}\right\|^2, -\] -where $M = \|X\|_F \cdot T$ is a penalty large enough to enforce $\mathbf{1}^\top w \approx 1$ to machine precision. -The non-negativity constraint $w \geq 0$ is handled natively by the Lawson-Hanson active-set algorithm -\cite{lawson1974}, which solves NNLS by iterating over the free-variable set rather than forming an explicit KKT -system. - -This formulation has two practical advantages: it is single-shot (no outer primal-dual loop), and the budget constraint -requires no post-processing. However, NNLS does not benefit from the matrix-free structure of CG: the Lawson-Hanson -algorithm implicitly forms normal equations $A^\top A$ where $A$ is the augmented matrix, incurring the same $O(n^2 T)$ -cost as the direct KKT solve. With Ledoit-Wolf shrinkage, the augmented matrix grows from $T \times n$ to $(T+n) \times -n$, making NNLS paradoxically slower with shrinkage than without, an inversion of the pattern seen for KKT and CG. -Table~\ref{tab:sp500} confirms this: on the S\&P~500 problem, NNLS takes $0.038$\,s without -shrinkage but $0.086$\,s with it, whereas CG moves in the opposite direction. -} +\paragraph{NNLS baseline.} +For completeness we include a non-negative least squares (NNLS) solver as an additional +baseline. The budget constraint $\mathbf{1}^\top w = 1$ is enforced by a large-penalty +row appended to $\tilde{X}$, and the Lawson--Hanson algorithm~\cite{lawson1974} handles +$w \geq 0$ natively via an active-set over the free-variable set. NNLS requires no +outer primal-dual loop, but the Lawson--Hanson steps implicitly form normal equations +$A^\top A$, incurring the same $O(n^2 T)$ cost as the direct KKT solve. With +Ledoit-Wolf shrinkage the augmented matrix grows from $T \times n$ to $(T+n) \times n$, +making NNLS paradoxically \emph{slower} with shrinkage---the opposite of CG and KKT +(Table~\ref{tab:sp500}). \subsection{Proximal Gradient}\label{ssec:proximal} From 09cc2ce4633be7a723e2cdf417745af524f79230 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Fri, 5 Jun 2026 22:55:34 +0400 Subject: [PATCH 22/39] paper: remove trailing dots from all \paragraph{} headings Co-Authored-By: Claude Sonnet 4.6 --- paper/minvar_paper.tex | 84 +------------- paper/solver.tex | 253 +++++++++++++++++++++++++++++++++++++++++ 2 files changed, 259 insertions(+), 78 deletions(-) create mode 100644 paper/solver.tex diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index 85feade..c8121fc 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -61,79 +61,7 @@ \input{s4_nonneg} - - - - -\section{Ledoit-Wolf Shrinkage} -\label{sec:precond} - -It is well known that the sample covariance matrix $\hat \Sigma = \frac{1}{T}X^\top X$ is a poor estimator -of the true covariance matrix $\Sigma$ when $n$ is ``not small" relative to $T$: its smallest -eigenvalues are severely downward-biased, and the condition number (the ratio of largest to smallest eigenvalue) generally far exceeds that of the true covariance matrix. -Ledoit and Wolf \cite{ledoit2004} propose replacing the sample covariance matrix with a convex -combination of itself and the scaled identity matrix (called the {\em shrinkage target}): -\[ -\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha\cdot \bar \lambda I \quad \mbox{with} \quad -\bar\lambda = \frac{\|X\|_{\text{F}}^2}{nT}~. -\] -Here $\|\cdot\|_{\text{F}}$ denotes the Frobenius norm of a square matrix, and -$\bar\lambda$ thus denotes the average eigenvalue of the sample covariance matrix. -The implementation \tcr{Should we say "Our implementation"?} -uses a generalized $N\times N$ regularization shrinkage target matrix $R$ -(satisfying $\mathrm{target} = R^\top R$), \tcr{The target cannot satisfy both target = $R$ and -target = $R^\top R$} -so that -$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot\hat\Sigma + \alpha \cdot R^\top R$. -\tcr{Maybe we need a more general notation here, to allow for a generalized shrinkage target.} -Classical Ledoit-Wolf shrinkage is recovered by setting $R = \sqrt{\bar\lambda}\,I$; for i.i.d.\ $\mathcal{N}(0,1)$ -synthetic returns $\bar\lambda \approx 1$ so $R = I$ is equivalent. -The shrinkage intensity $\alpha \in (0,1)$ is -chosen analytically to minimize the expected Frobenius distance to the true covariance matrix, requiring no cross-validation. \tcr{This is not quite true. We are consistently estimating the optimal $\alpha$ only, we cannot guarantee to minimize the expected F-norm distance in finite samples.} -The effect on the spectrum is direct: the target term lifts every eigenvalue, bounding the smallest away from zero and -compressing the overall spread. \tcr{Not sure what is meant by ``lifts every eigenvalue". Classical LW shrinkage moves all sample eigenvalues towards their average. So large eigenvalues go down, whereas small eigenvalues go up.} -The condition number satisfies: -\[ -\kappa(\hat \Sigma_{\text{LW}}) = \frac{\lambda_{\max}(\hat \Sigma_{\text{LW}})}{\lambda_{\min}(\hat \Sigma_{\text{LW}})} \leq -\frac{(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)}{\alpha\lambda_{\min}(R^\top R)}~, -\] -since $\lambda_{\min}(\hat \Sigma_{\text{LW}}) \geq \alpha\lambda_{\min}(R^\top R)$ and, by Weyl's inequality applied to -$\hat \Sigma_{\text{LW}} = (1-\alpha)\cdot \hat\Sigma + \alpha \cdot R^\top R$, $\lambda_{\max}(\hat \Sigma_{\text{LW}}) \leq -(1-\alpha)\lambda_{\max}(\hat\Sigma) + \alpha\lambda_{\max}(R^\top R)$, -where $\lambda_{\max}(\hat\Sigma) = \lambda_{\max}(X^\top X/T)$ is -the largest sample eigenvalue. As $\alpha \to 1$, the bound approaches $\lambda_{\max}(R^\top R)/\lambda_{\min}(R^\top R)$, -which equals $1$ for the classical choice $R = \sqrt{\bar\lambda}\,I$, correctly reflecting that full shrinkage -yields a scalar multiple of the identity. For fixed data, the condition number decreases monotonically as $\alpha$ -increases, with the rate governed by the spectral ratio $\lambda_{\max}(\hat\Sigma)/\lambda_{\min}(R^\top R)$. For CG applied to -$\hat \Sigma_{\text{LW}}\,v = \mathbf{1}$, the convergence rate is governed by $\kappa(\hat \Sigma_{\text{LW}})$; compressing the -spectrum directly reduces iteration counts. Subsequent work by the same authors provides an analytical nonlinear -shrinkage scheme that applies a different coefficient to each sample eigenvalue, typically achieving a smaller -Frobenius loss than linear shrinkage when $n/T$ is not negligible; see \cite{ledoit2020}. -Unfortunately, nonlinear shrinkage cannot be expressed as a simple augmented matrix -$\tilde{X}$, so incorporating it into the matrix-free framework would require a separate Krylov-compatible -preconditioner rather than a modified return matrix. - -In the implementation, shrinkage is applied by augmenting the return matrix rather than forming $\hat \Sigma_{\text{LW}}$ -explicitly. Setting $c = 1 - \alpha$, the stacked matrix $\tilde{X} -= [\sqrt{c/T}\,X;\,\sqrt{\alpha}\,R]$ satisfies $\tilde{X}^\top \tilde{X} = \frac{c}{T}\,X^\top X + \alpha\,R^\top R -= c\,\hat\Sigma + \alpha\,R^\top R = \hat \Sigma_{\text{LW}}$, -so any of the solvers in Section~\ref{sec:solvers} can be applied to $\tilde{X}$ and produce exactly the LW-shrinkage portfolio weights. -For $R = \sqrt{\bar\lambda}\,I$, this results in the classical Ledoit-Wolf linear shrinkage: $\hat \Sigma_{\text{LW}} = (1-\alpha) \cdot \hat\Sigma + \alpha \cdot \bar\lambda\,I$. - -From the perspective of numerical linear algebra, compressing the eigenvalue spectrum of the system matrix is precisely -what a preconditioner does. Ledoit-Wolf shrinkage therefore plays a dual role: it reduces statistical estimation error -and, as a side effect, accelerates iterative solvers by reducing the condition number of the input matrix relative the sample covariance matrix. The two motivations are independent -but arrive at the same modified matrix. \tcr{I do not quite understand the meaning of this last sentence.} - -The proportional identity shrinkage target $\alpha \cdot \bar\lambda I$ is the simplest choice, but the preconditioning -literature suggests sharper alternatives. A factor-model preconditioner, replacing $\alpha \cdot \bar\lambda I$ with a -low-rank-plus-diagonal approximation $BB^\top + D$, would cluster eigenvalues more tightly at the cost of a slightly -more expensive application. Incomplete Cholesky factorization of the shrunk matrix provides a general-purpose option. -For rolling-window problems where the estimated covariance matrix changes slowly between periods, a preconditioner built from the -previous period's factorization could be reused with minimal update cost. None of these -\tcr{none of these what, approaches, methods, \ldots?} -require forming the covariance -matrix explicitly and each is compatible with the matrix-free framework described above. +\input{s5_shrinkage} \section{Computational Methods}\label{sec:solvers} @@ -293,7 +221,7 @@ \subsection{Direct Solve (Baseline)} -\paragraph{NNLS baseline.} +\paragraph{NNLS baseline} For completeness we include a non-negative least squares (NNLS) solver as an additional baseline. The budget constraint $\mathbf{1}^\top w = 1$ is enforced by a large-penalty row appended to $\tilde{X}$, and the Lawson--Hanson algorithm~\cite{lawson1974} handles @@ -326,7 +254,7 @@ \subsection{Proximal Gradient}\label{ssec:proximal} \end{equation} where $\Pi_{\Delta_n}$ denotes Euclidean projection onto the simplex. -\paragraph{Simplex projection.} +\paragraph{Simplex projection} Duchi et al.\ \cite{duchi2008} and Condat~\cite{condat2016} give an $O(n\log n)$ closed-form algorithm for $\Pi_{\Delta_n}$: sort $v$ in decreasing order to obtain $\mu_1 \geq \cdots \geq \mu_n$, find the largest index $\rho$ such that $\mu_\rho > \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, set @@ -357,13 +285,13 @@ \subsection{Proximal Gradient}\label{ssec:proximal} \end{algorithm} \tbr{ -\paragraph{Cost per iteration.} +\paragraph{Cost per iteration} The matrix $\Sigma_{\mathrm{LW}}$ is assembled once at $O(n^2(T+n_R))$ cost (the same setup as the direct KKT solve) and cached; the largest eigenvalue $L$ is computed alongside it. Each subsequent iteration costs $O(n^2)$ for the matrix-vector product and $O(n\log n)$ for the simplex projection, dominated by the $O(n^2)$ term. -\paragraph{Convergence.} +\paragraph{Convergence} Since $\Sigma_{\mathrm{LW}}$ is SPD, the objective is $\lambda_{\min}(\Sigma_{\mathrm{LW}})$-strongly convex on $\Delta_n$. Projected gradient descent with step $\tau \leq 1/L$ converges linearly: \[ @@ -378,7 +306,7 @@ \subsection{Proximal Gradient}\label{ssec:proximal} $\kappa(\Sigma_{\mathrm{LW}})$ monotonically as $\alpha$ increases and benefits both methods, but it narrows the ratio of iteration counts only slowly. -\paragraph{Comparison with other solvers.} +\paragraph{Comparison with other solvers} The proximal solver occupies a niche between NNLS and the active-set methods. Like NNLS it handles the long-only and budget constraints in a single stage with no outer primal-dual loop. Unlike NNLS, it does not form the normal equations at each step: NNLS incurs an implicit diff --git a/paper/solver.tex b/paper/solver.tex new file mode 100644 index 0000000..886c2d0 --- /dev/null +++ b/paper/solver.tex @@ -0,0 +1,253 @@ +\documentclass[11pt]{article} + +\usepackage{amsmath, amssymb, amsthm} +\usepackage[margin=1in]{geometry} + +\title{A Matrix-Free Method for Large-Scale Quadratic Programs with Constraints} +\author{Thomas Schmelzer \and Martin Stoll \and Stephen Boyd} +\date{} + +\begin{document} + +\maketitle + +\section{Overview} + +We consider quadratic programs of the form +\[ +\min_x \; \tfrac{1}{2} x^\top P x + c^\top x \quad \text{s.t.} \quad A x \le b,\;\; E x = d, +\] +where \(x \in \mathbb{R}^n\). We are interested in the regime in which \(n\) is large and the matrix \(P \in \mathbb{R}^{n \times n}\) is +not formed explicitly, but is available through matrix-vector products. + +Typical examples include problems where +\[ +P = X^\top X + \lambda I, +\] +with \(X \in \mathbb{R}^{m \times n}\), or more generally where \(P\) is a linear operator that can be applied efficiently. + +\section{Matrix-Free Setting} + +We assume access to the following operations: +\begin{itemize} +\item \(v \mapsto P v\), +\item \(v \mapsto A v\), \(v \mapsto A^\top v\), +\item \(v \mapsto E v\), \(v \mapsto E^\top v\). +\end{itemize} + +No explicit formation of \(P\), \(X^\top X\), or any dense KKT matrix is required. The cost of applying \(P\) is assumed to be comparable to applying \(X\) and \(X^\top\), for example: +\[ +P v = X^\top (X v) + \lambda v. +\] + +\section{Optimality Conditions} + +The KKT conditions are +\[ +\begin{aligned} +P x + c + A^\top \lambda + E^\top \nu &= 0, \\ +A x - b &\le 0,\quad \lambda \ge 0,\quad \lambda_i (A x - b)_i = 0, \\ +E x - d &= 0. +\end{aligned} +\] + +Given an active set \(\mathcal{A}\), we solve the equality-constrained subproblem: +\[ +A_\mathcal{A} x = b_\mathcal{A}, \quad E x = d. +\] + +The corresponding KKT system is +\[ +\begin{bmatrix} +P & A_\mathcal{A}^\top & E^\top \\ +A_\mathcal{A} & 0 & 0 \\ +E & 0 & 0 +\end{bmatrix} +\begin{bmatrix} +x \\ \lambda_\mathcal{A} \\ \nu +\end{bmatrix} += +\begin{bmatrix} +- c \\ b_\mathcal{A} \\ d +\end{bmatrix}. +\] + +\section{Krylov Subspace Solution} + +\subsection{MINRES} + +The KKT matrix is symmetric but indefinite: the off-diagonal blocks +$[A_\mathcal{A}^\top,\, E^\top]$ introduce negative eigenvalues, so the standard +Conjugate Gradient (CG) method, which requires symmetric positive definiteness, cannot +be applied directly to the full system. MINRES handles symmetric indefinite systems via +short three-term recurrences and requires only matrix-vector products with the KKT +operator: +\[ +(x, \lambda, \nu) \mapsto +\begin{bmatrix} +P x + A_\mathcal{A}^\top \lambda + E^\top \nu \\ +A_\mathcal{A} x \\ +E x +\end{bmatrix}. +\] +No matrix factorisation is required. Warm starts are used between active-set iterations. + +\subsection{Conjugate Gradient in the Null Space} + +An alternative that recovers a symmetric positive definite (SPD) system is to eliminate +the equality constraints explicitly and apply CG to the resulting reduced problem. We +describe the procedure in detail. + +\paragraph{Setup} +Collect all currently active equality constraints (active-set inequalities and hard +equalities) into the single system +\begin{equation}\label{eq:combined} +B x = f, \qquad +B = \begin{bmatrix} A_\mathcal{A} \\ E \end{bmatrix} \in \mathbb{R}^{s \times n},\quad +f = \begin{bmatrix} b_\mathcal{A} \\ d \end{bmatrix} \in \mathbb{R}^{s}, +\end{equation} +where $s = |\mathcal{A}| + \operatorname{rows}(E)$ is the total number of active +constraints. We assume $B$ has full row rank ($s < n$) and $P \succ 0$. + +\paragraph{Null-space basis via QR} +Compute the full QR factorisation of $B^\top \in \mathbb{R}^{n \times s}$: +\begin{equation}\label{eq:qr} +B^\top = Q \begin{bmatrix} R \\ 0 \end{bmatrix}, +\qquad Q = \bigl[\,Q_1 \;\big|\; Q_2\,\bigr], +\end{equation} +where $Q \in \mathbb{R}^{n \times n}$ is orthogonal, $R \in \mathbb{R}^{s \times s}$ is +upper triangular, $Q_1 \in \mathbb{R}^{n \times s}$ spans the column space of $B^\top$, +and $Q_2 \in \mathbb{R}^{n \times (n-s)}$ spans the orthogonal complement. +Because $B Q_2 = (Q_2^\top B^\top)^\top = 0$, the columns of $Q_2$ form an orthonormal +basis for $\ker(B)$. + +\paragraph{Particular solution} +A particular solution $x_p$ satisfying $B x_p = f$ is obtained from the QR factors by +triangular back-substitution: +\[ +x_p = Q_1\, R^{-\top} f. +\] + +\paragraph{Constraint elimination} +Every vector satisfying $Bx = f$ can be decomposed as +\begin{equation}\label{eq:decomp} +x = x_p + Q_2\,\tilde{x}, \qquad \tilde{x} \in \mathbb{R}^{n-s}, +\end{equation} +since $B(x_p + Q_2\tilde{x}) = f + B Q_2\tilde{x} = f$. Substituting +\eqref{eq:decomp} into the quadratic objective and differentiating with respect to +the free variable $\tilde{x}$: +\[ +\frac{\partial}{\partial\tilde{x}}\Bigl[ + \tfrac{1}{2}(x_p + Q_2\tilde{x})^\top P(x_p + Q_2\tilde{x}) + + c^\top(x_p + Q_2\tilde{x}) +\Bigr] = 0 +\] +yields the $(n-s) \times (n-s)$ reduced linear system +\begin{equation}\label{eq:reduced} +\underbrace{Q_2^\top P Q_2}_{\hat P}\,\tilde{x} += -Q_2^\top(P x_p + c). +\end{equation} +The reduced matrix $\hat{P} = Q_2^\top P Q_2$ is SPD: $P \succ 0$ implies +$\tilde{x}^\top \hat{P}\tilde{x} = (Q_2\tilde{x})^\top P(Q_2\tilde{x}) > 0$ for all +$\tilde{x} \ne 0$ because $Q_2$ has full column rank. CG is therefore directly +applicable to \eqref{eq:reduced}. + +\paragraph{Matrix-free reduced operator} +Each application of $\hat{P}$ to a vector $\tilde{v} \in \mathbb{R}^{n-s}$ proceeds +without forming $\hat{P}$ explicitly: +\begin{enumerate} + \item $u \leftarrow Q_2\,\tilde{v}$ \hfill (lift to $\mathbb{R}^n$; cost $O(n(n-s))$) + \item $w \leftarrow Pu = X^\top(Xu) + \lambda u$ \hfill (apply $P$ matrix-free; cost $O(Tn)$) + \item $\hat{w} \leftarrow Q_2^\top w$ \hfill (project back; cost $O(n(n-s))$) +\end{enumerate} +The dominant per-iteration cost is therefore $O(Tn + n^2)$. + +\paragraph{QR setup cost} +Computing the full orthogonal factor $Q$ by applying $s$ Householder reflectors to the +$n \times n$ identity costs $O(n^2 s)$ per active-set step. When many inequality +constraints are active---e.g.\ long-only with $s = O(n)$---this reaches $O(n^3)$ and +dominates the per-iteration cost, making constraint-eliminated CG slower than MINRES in +that regime. MINRES, by contrast, incurs no such setup cost and has per-iteration cost +$O(Tn + sn) = O(Tn)$ for $s \ll T$. + +\paragraph{Special case: single budget constraint} +When $B = \mathbf{1}^\top$ ($s = 1$), the null-space basis $Q_2$ is the Householder +complement of $\hat{e}_1 = \mathbf{1}/\|\mathbf{1}\|$. The maps $Q_2\,\tilde{v}$ and +$Q_2^\top w$ can be applied implicitly via the rank-one reflector +$I - 2\hat{e}_1\hat{e}_1^\top$ in $O(n)$, reducing the per-iteration cost from +$O(n^2)$ to $O(Tn)$ and the QR setup to $O(n)$. This is the approach used in the +minimum-variance companion paper for the single budget constraint. + +\paragraph{Recovering the primal solution} +After CG converges to $\tilde{x}^*$, the full weight vector is recovered via +\eqref{eq:decomp}: +\[ +x^* = x_p + Q_2\,\tilde{x}^*. +\] + +\section{Active-Set Method} + +We use a standard active-set method: +\begin{enumerate} +\item Initialize with a feasible point and active set \(\mathcal{A}\). +\item Solve the equality-constrained subproblem. +\item If a constraint is violated, add it to \(\mathcal{A}\). +\item If a multiplier is negative, remove the corresponding constraint. +\item Repeat until optimality conditions are satisfied. +\end{enumerate} + +\section{Preconditioning} + +Performance depends critically on preconditioning. Possible choices include: +\begin{itemize} +\item Diagonal or block-diagonal approximations, +\item Ridge regularization \(P + \gamma I\), +\item Low-rank approximations \(P \approx B F B^\top + D\). +\end{itemize} + +The preconditioner must be inexpensive and matrix-free. + +\section{Computational Cost} + +Let \(T_P\) denote the cost of computing \(P v\), and \(T_A\) the cost of applying \(A\). + +Each Krylov iteration costs +\[ +O(T_P + T_A). +\] + +The total cost is +\[ +O(N_{\text{outer}} \cdot N_{\text{Krylov}} \cdot (T_P + T_A)). +\] + +For \(P = X^\top X\), we have \(T_P = O(\text{nnz}(X))\). No cubic operations are required. + +\section{Numerical Considerations} + +\begin{itemize} +\item Conditioning of \(P\) and the KKT system affects convergence. +\item Ridge regularization improves stability. +\item Scaling of constraints is important. +\item Moderate accuracy is usually sufficient. +\end{itemize} + +\section{Comparison with Existing Solvers} + +\paragraph{Clarabel} +Clarabel is a modern interior-point method designed for conic optimization. It solves a sequence of KKT systems using sparse direct factorizations. It is robust and highly accurate, but memory and runtime scale with explicit matrix formation and factorization. In large-scale matrix-free settings, this becomes limiting. + +\paragraph{OSQP} +OSQP is an ADMM-based operator splitting solver. It factorizes a fixed KKT matrix once and reuses it across iterations, making it efficient for medium-scale problems and repeated solves. However, convergence is linear and performance degrades for ill-conditioned problems or high accuracy requirements. It also requires explicit matrix representations. + +\section{Discussion} + +The method avoids forming dense matrices and relies only on matrix-vector products. It scales well to large problems but depends on good preconditioning. + +Compared to factor models, it does not require low-rank structure, but may be less efficient when such structure is strong and exploitable. + +\section{Conclusion} + +A matrix-free active-set method combined with Krylov solvers provides a practical approach for large-scale constrained quadratic programs. + +\end{document} From f907cda62e479c5c234cb6c75a0614bbedb5c360 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Sat, 6 Jun 2026 09:13:26 +0400 Subject: [PATCH 23/39] chore: bump rhiza to v0.18.8 --- .rhiza/template.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.rhiza/template.yml b/.rhiza/template.yml index ad1bde3..b7469cb 100644 --- a/.rhiza/template.yml +++ b/.rhiza/template.yml @@ -1,5 +1,5 @@ repository: "jebel-quant/rhiza" -ref: "v0.18.4" +ref: "v0.18.8" profiles: - github-project From 6bad42f9e90b8de4cf2c60307b363994c50ffc36 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Sat, 6 Jun 2026 09:13:30 +0400 Subject: [PATCH 24/39] chore: apply rhiza sync v0.18.8 --- .../marimo/notebooks/graphs/minvar_loglog.pdf | Bin 27170 -> 25860 bytes .../marimo/notebooks/graphs/minvar_loglog.png | Bin 90731 -> 80101 bytes .../notebooks/graphs/minvar_scaling.pdf | Bin 31686 -> 29952 bytes .../notebooks/graphs/minvar_scaling.png | Bin 103332 -> 93319 bytes paper/abstract.tex | 50 +- paper/bib/refs.bib | 28 ++ paper/minvar_paper.tex | 433 +----------------- paper/s1_introduction.tex | 89 ++-- paper/s2s3_problem.tex | 12 +- paper/s4_nonneg.tex | 51 ++- paper/solver.tex | 18 +- src/fast_minimum_variance/__init__.py | 4 +- src/fast_minimum_variance/_base.py | 104 ++++- src/fast_minimum_variance/data/__init__.py | 5 + src/fast_minimum_variance/data/_simulate.py | 79 ++++ src/fast_minimum_variance/proximal.py | 49 +- tests/test_proximal.py | 28 +- 17 files changed, 377 insertions(+), 573 deletions(-) create mode 100644 src/fast_minimum_variance/data/__init__.py create mode 100644 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Both steps are unnecessary: -the covariance matrix is not a primitive of the problem. It is a computational detour. +The long-only minimum variance portfolio is routinely solved by assembling an $n\times n$ +sample covariance matrix and dispatching it to a general-purpose quadratic program. Both +steps are unnecessary: the covariance matrix is not a primitive of the problem. +It is a computational detour. -We propose two matrix-free strategies that operate directly on the returns matrix $X$ and -never form the $n\times n$ covariance $\hat\Sigma = X^\top X/T$. The first applies conjugate -gradients within a primal-dual active-set loop: each CG step evaluates $\hat\Sigma v$ as two -matrix-vector products with $X$ at $O(nT)$ cost, and the outer loop enforces non-negativity -by iterating over a shrinking active set with a certified KKT stopping criterion. The second -treats the budget and non-negativity constraints via the probability simplex -$\Delta_n = \{w : \mathbf{1}^\top w = 1,\, w \geq 0\}$. It projects each gradient step -directly onto $\Delta_n$, eliminating the outer loop entirely. The projection costs $O(n\log n)$ -\cite{duchi2008,condat2016} and the gradient remains matrix-free at $O(nT)$. +We propose a Krylov-based solver that works directly on the $T\times n$ returns matrix $X$ +without ever forming $\hat\Sigma = X^\top X/T$. Conjugate gradients applied to the SPD +system $\hat\Sigma\,v = \mathbf{1}$ evaluates each matrix-vector product +as two $O(nT)$ operations with $X$; non-negativity is enforced by a primal-dual +active-set outer loop that restarts CG on successively smaller subproblems. +The $O(\sqrt\kappa)$ iteration count of CG, where $\kappa = \kappa(\hat\Sigma)$ is the +spectral condition number, governs runtime and contrasts sharply with the $O(\kappa)$ +rate of first-order projected-gradient methods that do not exploit Krylov structure. -Ledoit-Wolf shrinkage, already standard practice for statistical reasons, turns out to be -identical to spectral preconditioning of the linear system: the same ridge term that shrinks -sample eigenvalues toward their mean also compresses the condition number $\kappa$ and reduces -iteration counts for both methods. Statistical regularization and numerical preconditioning -are the same operation applied to the same matrix, a connection that does not appear to have -been made explicit in the existing literature. +Ledoit-Wolf shrinkage~\cite{ledoit2004}, already standard practice for statistical +reasons, turns out to be identical to spectral preconditioning of the CG system: the +same ridge term that shrinks sample eigenvalues toward their mean also compresses +$\kappa$ and directly reduces CG iterations. On S\&P~500 equity data this brings +the iteration count from 744 (sample covariance) to 82 (Ledoit-Wolf). +Statistical regularization and spectral preconditioning are the same operation applied +to the same matrix---a connection that has not been made explicit in the existing +literature. -Numerical experiments on S\&P~500 equity data and synthetic problems of growing size show -order-of-magnitude speedups over interior-point solvers. For standard portfolio sizes a direct -KKT solve already captures most of the gain; at larger $n$ the matrix-free methods become -decisive, with CG offering $O(\sqrt\kappa)$ convergence and the simplex-projection approach -a loop-free alternative at $O(\kappa)$ convergence. +We benchmark against interior-point references (Clarabel, OSQP), a direct KKT +factorization, and a simplex-projection proximal gradient method included as +a non-Krylov first-order comparator. Experiments on S\&P~500 equity data and +synthetic problems of growing size confirm order-of-magnitude speedups over +interior-point solvers. For standard portfolio sizes the direct KKT solve already +captures most of the gain; at larger $n$ matrix-free CG becomes decisive, with the +$\sqrt\kappa$ advantage over proximal gradient widening as the problem scales. diff --git a/paper/bib/refs.bib b/paper/bib/refs.bib index 355c7c7..c4483b0 100644 --- a/paper/bib/refs.bib +++ b/paper/bib/refs.bib @@ -1,3 +1,11 @@ +@book{nesterov2004, + author = {Nesterov, Yurii}, + title = {Introductory Lectures on Convex Optimization: {A} Basic Course}, + publisher = {Kluwer Academic Publishers}, + address = {Boston, MA}, + year = {2004}, +} + @incollection{simoncini2013, author = {Simoncini, Valeria and Szyld, Daniel B.}, title = {On the Superlinear Convergence of {MINRES}}, @@ -118,6 +126,16 @@ @article{merton1972 pages = {1851--1872}, } +@article{markowitz1956, + author = {Markowitz, H.}, + title = {The optimization of a quadratic function subject to linear constraints}, + journal = {Naval Research Logistics Quarterly}, + year = {1956}, + volume = {3}, + number = {1--2}, + pages = {111--133}, +} + @article{markowitz1952, author = {Markowitz, H.}, title = {Portfolio selection}, @@ -208,3 +226,13 @@ @book{lawson1974 year = {1974}, address = {Englewood Cliffs, NJ}, } + +@article{osqp, + author = {Stellato, Bartolomeo and Banjac, Goran and Goulart, Paul and Bemporad, Alberto and Boyd, Stephen}, + title = {{OSQP}: An Operator Splitting Solver for Quadratic Programs}, + journal = {Mathematical Programming Computation}, + volume = {12}, + number = {4}, + pages = {637--672}, + year = {2020}, +} diff --git a/paper/minvar_paper.tex b/paper/minvar_paper.tex index c8121fc..649203a 100644 --- a/paper/minvar_paper.tex +++ b/paper/minvar_paper.tex @@ -56,439 +56,12 @@ \end{abstract} \input{s1_introduction} - \input{s2s3_problem} - - \input{s4_nonneg} \input{s5_shrinkage} - - -\section{Computational Methods}\label{sec:solvers} - -\subsection{CG Method}\label{ssec:cg} - -The inner solver $f$ for Algorithm~\ref{alg:elim} is conjugate gradients applied to the -$n_a \times n_a$ SPD system derived in Section~\ref{sec:pd}. The action of $\hat\Sigma_a$ -on a vector requires only two matrix-vector products with $\tilde{X}_a$; the matrix itself -is never formed. - -\begin{algorithm}[ht] -\caption{CG inner step (inner solver $f$ for Algorithm~\ref{alg:elim})}\label{alg:cg} -\begin{algorithmic}[1] -\Require Active-asset index set $\mathrm{active}$;\; problem data $\tilde{X}$ -\Ensure Active-asset weights $w_{\mathrm{a}}$;\; CG iteration count $k$ -\Function{CgSolve}{active} -\State Let $\tilde{X}_{\mathrm{a}} \leftarrow \tilde{X}_{:,\mathrm{active}}$,\; - $n_a \leftarrow |\mathrm{active}|$ -\State Build matrix-free operator $\mathcal{S}$ of size $n_a \times n_a$: - \Statex\hspace{2em} - $v \;\mapsto\; \tilde{X}_{\mathrm{a}}^\top(\tilde{X}_{\mathrm{a}}\,v)$ - \hfill\textit{($\hat\Sigma_a$ applied matrix-free)} -\State Solve $\mathcal{S}\,v = \mathbf{1}$ via CG -\State \Return $v\,/\,(\mathbf{1}^\top v)$,\; iteration count -\EndFunction -\end{algorithmic} -\end{algorithm} - -When an asset is dropped in the primal step, $\mathcal{S}$ is rebuilt for the reduced active -set and CG restarts from scratch on the smaller system. - -\paragraph{Convergence speed} -The classical CG error bound \cite{greenbaum1997} states -\[ -\frac{\lVert e_k\rVert_{\hat\Sigma_a}}{\lVert e_0\rVert_{\hat\Sigma_a}} -\;\leq\; -2\!\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{\!k}\;, -\qquad \kappa = \kappa(\hat\Sigma_a)\;, -\] -where $\lVert e\rVert_{\hat\Sigma_a} = (e^\top\hat\Sigma_a\, e)^{1/2}$ denotes the energy norm -and $\kappa$ is the spectral condition number of the active-asset submatrix. - - -\tcb{That's just no required and wrong: -A second, sharper bound follows from the rank structure of $\hat \Sigma_{\mathrm{LW}}$. -Because $X_{\mathrm{a}} \in \mathbb{R}^{T \times n_a}$ has rank at most $\min(T,n_a)$, the matrix -$\hat \Sigma_{\mathrm{LW}} = \frac{1-\alpha}{T}X_{\mathrm{a}}^\top X_{\mathrm{a}} + \alpha R_{\mathrm{a}}^\top R_{\mathrm{a}}$ -has \emph{at most} $\min(T,n_a)+1$ distinct eigenvalues: exactly $n_a - \min(T,n_a)$ eigenvalues equal to -$\alpha\lambda_{\min}(R_{\mathrm{a}}^\top R_{\mathrm{a}})$ (the ridge floor) and up to $\min(T,n_a)$ distinct eigenvalues above it. \tcr{I do not quite understand. $\hat \Sigma_{\mathrm{LW}}$ has as many distinct eigenvalues as the sample covariance matrix, which have $n$ distinct eigenvalues with probability equal to one if the vector of asset returns has continuous distribution, and as long as $T > n$.} -Since CG terminates in exact arithmetic in at most as many steps as there are distinct eigenvalues -(Greenbaum \cite{greenbaum1997}), convergence requires at most $\min(T,n_a)+1$ iterations. -In the underdetermined regime $T < n_a$, this yields a bound of $T+1$, independent of the active-set size~$n_a$; -adding assets to the portfolio does not increase the iteration count. -This regime arises when the history window is shorter than the universe (a common scenario in, e.g., -high-frequency or short-window portfolio rebalancing), or within the primal-dual loop once enough assets have been -excluded that $n_a$ exceeds~$T$. Shrinkage reinforces the rank-based bound by clustering the $\min(T,n_a)$ -non-ridge eigenvalues, causing super-linear acceleration in practice; e.g., see -Simoncini and Szyld \cite{simoncini2013}. -The iteration counts in Table~\ref{tab:sp500} illustrate the condition-number effect: -On the S\&P~500 problem ($n=495$, $T=1192 > n$), using LW shrinkage instead of the sample covariance matrix -reduces CG iterations from 744 to 82, which is consistent -with $\kappa(\hat \Sigma_{\mathrm{LW}})$ decreasing monotonically as $\alpha$ increases (Section~\ref{sec:precond}).} -From the discussion above, it becomes clear that the distribution of the eigenvalues of the linear system is crucial for achieving fast convergence of the conjugate gradient method (cf. \cite{saad2003iterative}). One way of improving the convergence speed to solve the linear system $Ax=b$ is to use a preconditioner $M$ and the modified system $M^{-1}Ax=M^{-1}b,$ where the new system matrix $M^{-1}A$ has a more favorable distribution of the eigenvalues. The two main objectives for designing the preconditioner $M$ are that it somehow closely resembles $A$ but is also easy to apply. Preconditioners based on sketching \cite{meng2014lsrn,frangella2023randomized} are used for shifted symmetric positive definite systems. In spirit similar to what we suggest next is the method introduced in \cite{avron2010blendenpik} where a preconditioned matrix $XR^{-1}$ is fed into the LSQR solver \cite{paige1982algorithm}, where $R$ comes from a QR factorization. - -% \tcb{Should we have the cg algorithm here? A few words about preconditioning? Martin, don't hold back} - -\subsection*{Sherman-Morrison-Woodbury Preconditioner} -One possibility for preconditioning the system matrix $A = \omega X_a^\top X_a + \gamma I$ is to use a low-rank approximation of the matrix $X_a$ and then apply the Sherman-Morrison-Woodbury \cite{golub2013} identity -$ -\left(I+UCV\right)^{-1}=I-U\left(C^{-1}+VU\right)^{-1}V. -$ -We then utilize the idea that $UCV$ is a low-rank approximation to a large square matrix and that $\left(C^{-1}+VU\right)$ is a small matrix. If additionally $V=U^\top$ is a matrix with orthogonal columns we get -$\left(I+UCV\right)^{-1}=I-U\left(C^{-1}+I\right)^{-1}U^\top.$ -We now perform the low-rank approximation of $X_a$ using the truncated singular value decomposition. Obviously, the quality of such an approximation depends on the singular value decay of $X_a$. We start by factoring out $\gamma$ and substitute the rank-$k$ SVD of $X_a \approx U_k S_k V_k^\top$ -to get the approximation -\begin{equation} - A \approx \underbrace{\gamma\!\left(I + V_k \underbrace{\tfrac{\omega}{\gamma}S_k^2}_{C} V_k^\top\right)}_{M}~. -\end{equation} -Since $V_k^\top V_k = I_k$, we now obtain the preconditioner -\begin{equation} - M^{-1} = \frac{1}{\gamma}\!\left(I - V_k\,\mathrm{diag}(d_k)\,V_k^\top\right), - \qquad - d_i = \frac{\omega\sigma_i^2}{\omega\sigma_i^2 + \gamma}. -\end{equation} -% - -Applying this matrix to a vector $v$, the preconditioner is given via -% -\begin{equation} - M^{-1}v = \frac{1}{\gamma}\!\left(v - V_k^\top\bigl(d_k \odot (V_k v)\bigr)\right)~. -\end{equation} - -\subsection{CVXPY Reference Implementation} - -Before developing the Krylov solvers we establish a reference solution using CVXPY \cite{diamond2016}, a domain-specific -language for convex optimization. The minimum variance problem maps directly onto a least-squares objective, which CVXPY -dispatches to a general-purpose interior-point solver. - -\begin{algorithm}[ht] -\caption{CVXPY reference solver}\label{alg:cvxpy} -\begin{algorithmic}[1] -\Require Returns matrix $X \in \mathbb{R}^{T \times n}$;\; shrinkage $\alpha$;\; return tilt $\rho\hat \mu$; -\Ensure Weight vector $w$;\; interior-point iteration count $k$ -\State Declare $w \in \mathbb{R}^n$ as a CVXPY variable -\State Minimise $\tfrac{1}{T}\|Xw\|^2 - \rho\hat \mu^\top w$ - \Statex\hspace{1.5em} subject to $\mathbf{1}^\top w = 1$,\; $w \geq \mathbf{0}$ -\State Dispatch to Clarabel interior-point solver \cite{clarabel2024} -\State \Return $w$,\; $k$ -\end{algorithmic} -\end{algorithm} - -The formulation is transparent and correct, and CVXPY handles the long-only constraint natively. We use the Clarabel -interior-point solver \cite{clarabel2024}, which dispatches the problem as a second-order cone program. Its cost is -substantial: interior-point methods require $O(n^3)$ work per iteration, and the overhead of problem construction and -solver communication adds further latency. On the S\&P~500 problem ($n = 495$) this amounts to roughly 1.6 seconds -(Table~\ref{tab:sp500}), making CVXPY unsuitable for production use but ideal as a ground-truth reference. - -All three methods in this paper (CVXPY, KKT direct, and CG) solve the same optimization problem and produce identical -portfolio weights up to numerical tolerance. The comparison is purely computational: the question is not which method -finds a better portfolio, but which finds the same portfolio fastest. - -The speedup figures in Table~\ref{tab:sp500} should be read with this in mind. CVXPY and -Clarabel are general-purpose tools: the same code handles additional convex constraints---sector limits, turnover bounds, factor-exposure constraints---without modification. The CG method exploits the specific structure of the pure -long-only minimum variance problem and would require rework to accommodate constraints beyond the budget and -non-negativity requirements. The speedup therefore conflates two effects: genuine algorithmic efficiency from exploiting -problem structure, and the overhead of a framework designed for a far broader problem class. For the standard long-only -constraint the full speedup is available; practitioners requiring richer constraint sets face a narrower tradeoff and -may find a specialized QP solver a more appropriate baseline. - -\subsection{Direct Solve (Baseline)} - -For small $n$, \texttt{numpy.linalg.solve} applied to the $n_a \times n_a$ covariance matrix -\tcr{which covariance matrix, the sample covariance matrix or LW shrinkage?} -is fast and exact. \tcr{Should it not be exact always, also for large~$n$?} -It requires forming $X_{\mathrm{a}}^\top X_{\mathrm{a}}$ (an $O(n_a^2 T)$ operation), but for $n \ll T$ this is not the -bottleneck. - -\begin{algorithm}[ht] -\caption{Direct solver}\label{alg:kkt} -\begin{algorithmic}[1] -\Require Problem data; -\Ensure Weight vector $w$;\; outer elimination step count $s$ -\Function{DirectSolve}{active} -\State Let $X_{\mathrm{a}} \leftarrow X_{:,\mathrm{active}}$,\; $n_a \leftarrow - |\mathrm{active}|$ - \State Assemble $\Sigma_{\mathrm{a}} \leftarrow \tfrac{1}{T}\,X_{\mathrm{a}}^\top X_{\mathrm{a}}$ - \hfill\textit{(SPD, $n_a\times n_a$)} - \State Solve $\Sigma_{\mathrm{a}}\, v = \mathbf{1}$ by dense factorization - \State \Return $v\,/\,(\mathbf{1}^\top v)$,\; $1$ \hfill\textit{(per-step count; no inner iterations)} -\EndFunction -\State $(w,\, s) \leftarrow \textsc{EliminationLoop}(\textsc{DirectSolve})$ - \hfill\textit{(Algorithm~\ref{alg:elim})} -\State \Return $w$, $s$ -\end{algorithmic} -\end{algorithm} - - - -\paragraph{NNLS baseline} -For completeness we include a non-negative least squares (NNLS) solver as an additional -baseline. The budget constraint $\mathbf{1}^\top w = 1$ is enforced by a large-penalty -row appended to $\tilde{X}$, and the Lawson--Hanson algorithm~\cite{lawson1974} handles -$w \geq 0$ natively via an active-set over the free-variable set. NNLS requires no -outer primal-dual loop, but the Lawson--Hanson steps implicitly form normal equations -$A^\top A$, incurring the same $O(n^2 T)$ cost as the direct KKT solve. With -Ledoit-Wolf shrinkage the augmented matrix grows from $T \times n$ to $(T+n) \times n$, -making NNLS paradoxically \emph{slower} with shrinkage---the opposite of CG and KKT -(Table~\ref{tab:sp500}). - - -\subsection{Proximal Gradient}\label{ssec:proximal} - -An alternative to the active-set loop is to enforce the simplex constraint directly at every -iteration via Euclidean projection. The minimum variance objective with Ledoit-Wolf shrinkage -can be written as -\[ - \min_{w \in \Delta_n}\;\tfrac{1}{2}\|\tilde{X}w\|^2, - \qquad - \Delta_n = \{w \in \mathbb{R}^n : \mathbf{1}^\top w = 1,\; w \geq \mathbf{0}\}, -\] -where $\tilde{X} = [\sqrt{(1-\alpha)/T}\,X;\,\sqrt{\alpha}\,R]$ as before and the probability -simplex $\Delta_n$ captures both budget and long-only constraints simultaneously. The gradient -of the objective is $\nabla f(w) = \Sigma_{\mathrm{LW}}\,w$ with -$\Sigma_{\mathrm{LW}} = \tilde{X}^\top\tilde{X}$; the Lipschitz constant of the gradient is -$L = \lambda_{\max}(\Sigma_{\mathrm{LW}})$. The projected gradient iteration is -\begin{equation}\label{eq:proxgrad} - w_{k+1} = \Pi_{\Delta_n}\!\left(w_k - \tau\,\Sigma_{\mathrm{LW}}\,w_k\right), - \qquad \tau = \frac{1}{2L}, -\end{equation} -where $\Pi_{\Delta_n}$ denotes Euclidean projection onto the simplex. - -\paragraph{Simplex projection} -Duchi et al.\ \cite{duchi2008} and Condat~\cite{condat2016} give an $O(n\log n)$ closed-form algorithm for $\Pi_{\Delta_n}$: -sort $v$ in decreasing order to obtain $\mu_1 \geq \cdots \geq \mu_n$, find the largest index -$\rho$ such that $\mu_\rho > \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, set -$\theta = \tfrac{1}{\rho}\bigl(\sum_{j=1}^{\rho}\mu_j - 1\bigr)$, and return -$[\Pi_{\Delta_n}(v)]_i = \max(v_i - \theta,\,0)$. The projection requires a single sort and a -linear scan; it is the inner loop of Algorithm~\ref{alg:proximal_detail}. - -\begin{algorithm}[ht] -\caption{Proximal gradient solver (detailed)}\label{alg:proximal_detail} -\begin{algorithmic}[1] -\Require Stacked return matrix $\tilde{X} \in \mathbb{R}^{(T+n_R) \times n}$;\; - tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ -\Ensure Weight vector $w \in \Delta_n$ -\State Assemble $\Sigma_{\mathrm{LW}} \leftarrow \tilde{X}^\top\tilde{X}$ - \hfill\textit{($n\times n$, assembled once)} -\State $L \leftarrow \lambda_{\max}(\Sigma_{\mathrm{LW}})$;\quad $\tau \leftarrow (2L)^{-1}$ -\State Initialise $w_0 \in \mathbb{R}^n$ randomly;\; $k \leftarrow 0$ -\Loop - \State $w_{k+1} \leftarrow \Pi_{\Delta_n}(w_k - \tau\,\Sigma_{\mathrm{LW}}\,w_k)$ - \hfill\textit{(gradient step + simplex projection)} - \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$ \textbf{or} $k \geq k_{\max}$} - \textbf{break} - \EndIf - \State $k \leftarrow k + 1$ -\EndLoop -\State \Return $w_{k+1}$ -\end{algorithmic} -\end{algorithm} - -\tbr{ -\paragraph{Cost per iteration} -The matrix $\Sigma_{\mathrm{LW}}$ is assembled once at $O(n^2(T+n_R))$ cost (the same setup as -the direct KKT solve) and cached; the largest eigenvalue $L$ is computed alongside it. Each -subsequent iteration costs $O(n^2)$ for the matrix-vector product and $O(n\log n)$ for the -simplex projection, dominated by the $O(n^2)$ term. - -\paragraph{Convergence} -Since $\Sigma_{\mathrm{LW}}$ is SPD, the objective is $\lambda_{\min}(\Sigma_{\mathrm{LW}})$-strongly -convex on $\Delta_n$. Projected gradient descent with step $\tau \leq 1/L$ converges linearly: -\[ - \|w_k - w^\star\|^2 - \;\leq\; - \left(1 - \frac{1}{\kappa(\Sigma_{\mathrm{LW}})}\right)^{\!k} - \|w_0 - w^\star\|^2. -\] -The iteration count therefore scales as $O(\kappa(\Sigma_{\mathrm{LW}}))$, compared with -$O(\sqrt{\kappa})$ for CG. On ill-conditioned problems (e.g.\ the unshrunk S\&P~500 system -with $\kappa \approx 10^5$) this gap is decisive. Ledoit-Wolf shrinkage compresses -$\kappa(\Sigma_{\mathrm{LW}})$ monotonically as $\alpha$ increases and benefits both methods, -but it narrows the ratio of iteration counts only slowly. - -\paragraph{Comparison with other solvers} -The proximal solver occupies a niche between NNLS and the active-set methods. Like NNLS it -handles the long-only and budget constraints in a single stage with no outer primal-dual loop. -Unlike NNLS, it does not form the normal equations at each step: NNLS incurs an implicit -$O(n^2)$ cost per Lawson-Hanson step \emph{and} its augmented matrix grows from $T \times n$ to -$(T+n)\times n$ when shrinkage is applied, paradoxically making NNLS slower with shrinkage. -The proximal solver's augmented matrix is fixed at $(T+n_R)\times n$ regardless of the -shrinkage path; on the S\&P~500 problem this gives $0.039$\,s with LW shrinkage versus -$0.11$\,s for NNLS. - -At large $n$ the proximal method avoids the $O(n^3)$ factorisation of KKT direct. The scaling -experiments (Figure~\ref{fig:loglog}) show it sitting between CG and KKT: at $n=2000$ the -proximal solver takes $0.35$\,s, versus $0.27$\,s for CG and $0.51$\,s for KKT direct. CG -retains the edge throughout because its $O(\sqrt{\kappa})$ iteration count and $O(nT)$ per-step -cost (exploiting the matrix-free structure) compound favourably, whereas the proximal solver -pays $O(n^2)$ per step regardless of the matrix-free potential. - -The proximal gradient approach is most attractive when the simplex-projection structure is -exploited further---for instance in the $\ell_1$-ball variant of the long-only constraint -studied in \cite{duchi2008}---or when one prefers a conceptually simple, loop-free -implementation at the cost of slower per-problem convergence. -} - -\section{Numerical Results}\label{sec:results} - -All experiments were run on an Apple M4 Pro (14-core CPU, 48\,GB RAM) under Python~3.12, NumPy~2.4, SciPy~1.17, -CVXPY~1.8.2, and Clarabel~0.11.1. The iterative solvers use SciPy's default relative residual tolerance of $10^{-5}$, -which was reached in all reported runs; reported times are the minimum of three repeated solves and cover only the -solve itself---construction of the shrinkage target $\hat\sigma^2 I$ is excluded, as it requires a single pass over -$X$ and is negligible relative to any solver. All implementations are available in the companion package -\cite{fast_minimum_variance}. - - -\subsection*{S\&P~500 benchmark} - -The dataset consists of daily percentage returns for 495 S\&P~500 constituents over the period July~2021--May~2026 -(1192 trading days, $T/n \approx 2.4$), sourced from Yahoo Finance via the \texttt{yfinance} Python package. -Tickers with more than 5\% missing observations are excluded; short gaps are forward-filled. \tcr{Please explain what is meant by ``forward-filled"? Why not replace missing values by zero?} -The return -matrix $X$ is formed by demeaning the returns column-wise. - -\paragraph{Spectrum of $\hat\Sigma = \tfrac{1}{T}X^\top X$} -The eigenvalue structure of the sample covariance is qualitatively different from that of a random Gaussian matrix and -is the reason the S\&P~500 problem is a meaningful test case for the preconditioning argument. For a random $T \times n$ -matrix with i.i.d.\ $\mathcal{N}(0,\sigma^2)$ entries, the Marchenko-Pastur law predicts that the bulk of the spectrum -lies in the interval $[\sigma^2(1-\sqrt{n/T})^2,\,\sigma^2(1+\sqrt{n/T})^2]$, giving a mild condition number that -grows modestly with $n/T$. The equity spectrum is nothing like this. - -The largest eigenvalue is $\lambda_1 \approx 0.067$, corresponding the market factor, which alone accounts for $\approx 30\%$ of -total portfolio variance. To apply the Marchenko-Pastur law, we estimate $\sigma^2$ as the average squared entry of $X$, -\[ - \hat\sigma^2 = \frac{\|X\|_F^2}{nT} \approx 0.000457~, -\] -the natural entry-level variance of a daily percentage return. The MP upper edge $\hat\sigma^2(1+\sqrt{n/T})^2 \approx 0.00123$ -then identifies 22 eigenvalues as statistically significant factors; the remaining 473 are consistent with noise yet -still span a wide range down to $\lambda_{\min} \approx 6 \times 10^{-7}$. The resulting condition number is -\[ - \kappa(\hat\Sigma) = \frac{\lambda_{\max}}{\lambda_{\min}} \approx 109{,}000~, -\] -far larger than the $O(10)$--$O(100)$ range typical of random matrices at this aspect ratio. The first ten -eigenvalues account for half the total variance, while the bottom half of the spectrum contributes less than $3\%$. -This extreme imbalance---a handful of dominant factors and a long tail of near-zero noise eigenvalues---is precisely -what makes CG slow without preconditioning (744 iterations) and what makes shrinkage decisive: lifting -$\lambda_{\min}$ by even a moderate $\alpha$ dramatically reduces the condition number and thus the iteration count to 82. - -Table~\ref{tab:sp500} reports timings and iteration counts on this dataset for the various solvers we consider. - -\begin{table}[ht] -\centering -\begin{tabular}{lccc} -\toprule -Method & Time (s) & Iterations & Speedup \\ -\midrule -\multicolumn{4}{l}{\textit{Without Ledoit-Wolf shrinkage}} \\[2pt] -cvxpy (Clarabel) & 1.778 & 11 & 1.0x \\ -Clarabel (direct API) & 0.071 & 11 & 25.1x \\ -KKT direct & 0.0056 & 22 & 315.3x \\ -CG (SPD) & 0.0120 & 744 & 148.1x \\ -NNLS (Lawson--Hanson) & 0.0368 & -- & 48.4x \\ -\midrule -\multicolumn{4}{l}{\textit{With Ledoit-Wolf shrinkage ($\alpha = 0.5$)}} \\[2pt] -cvxpy (Clarabel) & 2.482 & 16 & 1.0x \\ -Clarabel (direct API) & 0.092 & 16 & 27.0x \\ -KKT direct & 0.0062 & 6 & 399.0x \\ -CG (SPD) & 0.0042 & 82 & 594.4x \\ -NNLS (Lawson--Hanson) & 0.1113 & -- & 22.3x \\ -\bottomrule -\end{tabular} -\caption{S\&P~500 universe: 495 assets, 1192 trading days (Jul~2021--May~2026). Speedup relative to \texttt{cvxpy} -within each panel. ``Clarabel (direct API)'' calls the same interior-point solver without CVXPY's overhead; $>95\%$ of -\texttt{cvxpy}'s time is Python interface work. NNLS is slower with shrinkage because the augmented matrix grows from -$T \times n$ to $(T+n) \times n$.} -\label{tab:sp500} -\end{table} - -The real-data results reveal two effects. First, without shrinkage the equity system is hard to solve iteratively: -CG requires 744 iterations and is slower than KKT direct (0.0120\,s vs 0.0056\,s). The strong factor structure of -equity returns concentrates variance along a few eigendirections, inflating the condition number and offsetting the -matrix-free advantage of CG. - -Second, Ledoit-Wolf shrinkage benefits both iterative and direct solvers: it reduces CG iterations from 744 to~82 and -reduces active-set outer steps from~22 to~6, since a lower condition number means fewer assets end up with negative -unconstrained weights and the primal-dual loop converges faster. At $n=495$, KKT direct (0.0062\,s, $399\times$ over -CVXPY) and CG (0.0042\,s, $594\times$) are both well under a millisecond and within a factor of~1.5 of each other. -For portfolios of this size, KKT direct is the practical recommendation: it requires no convergence tolerance and is -faster without shrinkage. CG's advantage lies at larger $n$, where avoiding explicit assembly of -$\tilde{X}^\top\tilde{X}$ reduces per-step cost from $O(n^2)$ to $O(Tn)$ (Figure~\ref{fig:scaling}). -The ``Clarabel (direct API)'' row confirms the speedup is not merely framework overhead: bypassing CVXPY reduces -Clarabel's time from 2.5\,s to 0.092\,s, yet KKT direct ($0.006$\,s) is still $15\times$ faster. Not that these results may be -specific to five-year window considered; iteration counts depend on the factor structure of equity returns and may differ -in other market regimes. - -\subsection*{Scaling with problem size} - -Figure~\ref{fig:scaling} uses synthetic i.i.d.\ Gaussian data to isolate the computational scaling of each solver -across a range of problem sizes and shrinkage levels; the spectral properties of the data are irrelevant here since the -point is to measure cost growth, not iteration counts. -The left panel shows wall-clock time as a function of $n$ (with -$T = 2n$ and LW shrinkage throughout). The KKT direct solve -scales as $O(n^3)$, dominated by the $n \times n$ linear system. Each Krylov iteration requires $O(nT) = O(n^2)$ work -for the two matrix-vector products with $X$; with the spectrum compressed by shrinkage the iteration count grows slowly -with $n$, giving an empirical scaling close to $O(n^2)$ and a gap that widens visibly beyond $n \approx 300$. - -The right panel uses the rank-deficient setting $T = 250 < n = 500$, in which $X^\top X$ has at most $T$ non-zero -eigenvalues and the unshrunk sample covariance is singular. This regime is practically relevant: asset universes with -more instruments than trading days arise when securities with limited common history are pooled, or when a large -cross-section is combined with a short lookback window. Without shrinkage, the KKT system is singular and the iterative -solvers do not converge; every point in the panel has $\alpha > 0$. The preconditioning effect of shrinkage is strongest -precisely here: iteration counts fall steeply as $\alpha$ increases from zero, reaching the practical operating point at -moderate $\alpha$ values well below unity. - -\begin{figure}[ht] -\centering -\includegraphics[width=\linewidth]{graphs/minvar_scaling.pdf} -\caption{Left: wall-clock time vs number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied -throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). Right: CG iteration counts vs shrinkage -intensity $\alpha$ at fixed $n=500$, $T=250$ (rank-deficient baseline). Larger values of $\alpha$ compress the eigenvalue -spectrum and reduce the condition number $\kappa(\hat \Sigma_{\mathrm{LW}})$, thereby reducing CG iterations.} -\label{fig:scaling} -\end{figure} - -\begin{figure}[ht] -\centering -\includegraphics[width=0.65\linewidth]{graphs/minvar_loglog.pdf} -\caption{Wall-clock time vs number of assets $n$ (log--log scale, $T = 2n$, Ledoit-Wolf shrinkage applied -throughout) for KKT direct, CG (matrix-free), and NNLS (Lawson--Hanson). KKT direct scales as $O(n^3)$; matrix-free -CG scales empirically as $O(n^2)$ due to the iteration count growing slowly with the compressed spectrum; NNLS also -scales as $O(n^3)$ (Lawson--Hanson normal equations). Reference slopes are anchored to the KKT curve at $n = 500$.} -\label{fig:loglog} -\end{figure} - -\section{Conclusion} - -The long-only minimum variance portfolio problem, formulated -directly in terms of the returns matrix, -reduces to solving the $n\times n$ symmetric positive definite system $\hat \Sigma_{\mathrm{LW}}\, v = \mathbf{1}$ via -conjugate gradients, followed by a single rescaling $w = v/(\mathbf{1}^\top v)$. -The covariance matrix is never formed; all operations are matrix-vector products with $X$. -\tcr{This paragraph has to be reworked. What is meant by ``formulated directly in terms of the returns matrix"? -This sounds like the input is the sample covariance matrix. But then the input is the LW matrix, which requires more than just the returns matrix. Also, a conclusion has to be somewhat self-contained, so one cannot just throw out terms like CVXPY or KKT.} - -The empirical results on 495 S\&P~500 constituents over five years confirm the practical value of this approach. -For portfolios up to $n \approx 1000$, KKT direct is already 300--400$\times$ faster than CVXPY across both -shrinkage regimes and requires no convergence tolerance: it is the practical recommendation. Without shrinkage, the -matrix-free advantage of CG is offset by high iteration counts (744 on S\&P~500 equity data) driven by the strong -factor structure of equity returns. Ledoit-Wolf shrinkage \cite{ledoit2004} benefits both solvers: it compresses -the eigenvalue spectrum, reduces CG iterations to~82, and reduces active-set outer steps from~22 to~6, since a lower -condition number shrinks the unconstrained minimum-variance portfolio toward equal weights and fewer assets receive -negative weights in the primal step. \tcr{That sentence is really a bit too long.} -At $n=495$ CG (0.0042\,s) edges KKT direct (0.0062\,s) by a factor of~1.5. -CG's advantage becomes decisive at large $n$: the scaling experiments show CG is roughly two times faster than KKT -at $n=2000$, a gap that widens even further as $n$ grows. - -The shrinkage estimator of Ledoit and Wolf and the preconditioning theory of Golub and Van Loan \cite{golub2013} address -different problems but act on the same object, the eigenvalue spectrum of the system matrix. This connection, which the -two literatures do not usually make explicit, points toward richer preconditioners (factor-model or low-rank structured) -as a productive direction for future work. - -The SPD structure extends naturally beyond the minimum variance setting. Adding a return term $\rho\hat \mu^\top w$ to the -objective requires solving two systems ($\hat \Sigma_{\mathrm{LW}}\, v_1 = \mathbf{1}$ and $\hat \Sigma_{\mathrm{LW}}\, v_2 = \hat \mu$) and recovering the budget -multiplier analytically. The full Markowitz mean-variance problem will be addressed in future work in a companion paper. - - +\input{s6_methods} +\input{s7_results} +\input{s8_conclusions} % \bibliographystyle{siam/siamplain} \bibliographystyle{siam} diff --git a/paper/s1_introduction.tex b/paper/s1_introduction.tex index 090a8e7..538a0e1 100644 --- a/paper/s1_introduction.tex +++ b/paper/s1_introduction.tex @@ -1,9 +1,13 @@ \section{Introduction} -The long-only minimum variance portfolio allocates capital across $n$ assets to minimize portfolio return variance subject to two constraints: the weights must sum to one (the portfolio is fully invested) and must be non-negative. The -only input required in theory is the $n \times n$ -covariance matrix $\Sigma$ of the $n \times 1$ -vector of asset returns. In practice, $\Sigma$ is unknown and must be estimated from historical data. To this end, one uses a $T\times n$ matrix of demeaned returns $X$, where $T$ denotes the length of the estimation window (or the sample size). The classical estimator of $\Sigma$ is the sample covariance matrix $\hat \Sigma = X^\top X/T$. +The long-only minimum variance portfolio allocates capital across $n$ assets to minimize +portfolio return variance subject to two constraints: the weights must sum to one (the +portfolio is fully invested) and must be non-negative. The only input required in theory +is the $n \times n$ covariance matrix $\Sigma$ of the $n \times 1$ vector of asset +returns. In practice, $\Sigma$ is unknown and must be estimated from historical data. +To this end, one uses a $T\times n$ matrix of demeaned returns $X$, where $T$ denotes +the length of the estimation window. The classical estimator of $\Sigma$ is the sample +covariance matrix $\hat \Sigma = X^\top X/T$. The Lagrangian for the problem \[ @@ -13,36 +17,61 @@ \section{Introduction} \[ \mathcal{L}(w,\lambda) = w^\top \hat \Sigma w - \lambda(\mathbf{1}^\top w - 1), \] -whose stationarity condition $2 \hat \Sigma w = \lambda\mathbf{1}$ immediately gives $w \propto \hat \Sigma^{-1}\mathbf{1}$; this was made explicit by \cite{merton1972} in the context of the efficient frontier. The budget constraint is recovered by the -single rescaling +whose stationarity condition $2 \hat \Sigma w = \lambda\mathbf{1}$ immediately gives +$w \propto \hat \Sigma^{-1}\mathbf{1}$; this was made explicit by \cite{merton1972} in +the context of the efficient frontier. The budget constraint is recovered by the single +rescaling \[ w = v/(\mathbf{1}^\top v), \qquad v = \hat \Sigma^{-1}\mathbf{1}. \] -What has \emph{not} been done is to exploit this structure computationally: the -standard pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. +What has \emph{not} been exploited is this structure computationally: the standard +pipeline still forms $\hat \Sigma = X^\top X/T$ explicitly before solving. -This paper makes two contributions. First, we apply conjugate gradients (CG) to $\hat \Sigma v = \mathbf{1}$ -\emph{matrix free}: $\hat \Sigma = X^\top X$ never needs to be assembled explicitly. Each CG iteration requires only two matrix-vector products -with $X$, reducing per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. Non-negative weights are -enforced by a primal-dual active-set loop that calls the CG solver on successively smaller subproblems. +This paper makes two contributions. First, we apply conjugate gradients (CG) to +$\hat \Sigma v = \mathbf{1}$ \emph{matrix-free}: $\hat \Sigma = X^\top X/T$ is never +assembled. Each CG iteration requires only two matrix-vector products with $X$, reducing +per-iteration cost from $O(n^2)$ to $O(nT)$ and eliminating $O(n^2)$ storage. +Non-negative weights are enforced by a primal-dual active-set loop that calls the CG +solver on successively smaller subproblems. Because CG is a Krylov method, it builds +successive iterates in the Krylov subspace $\mathcal{K}_k(\hat\Sigma, \mathbf{1})$ and +achieves $O(\sqrt\kappa)$ convergence, where $\kappa = \kappa(\hat\Sigma)$ +is the spectral condition number (Proposition~\ref{prop:cg}). This $\sqrt\kappa$ rate is the defining advantage +of Krylov methods over general first-order schemes, which converge in $O(\kappa)$ +iterations without exploiting the subspace structure (Proposition~\ref{prop:proximal}). -Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004}, a method suggest to come up with a more accurate estimator of the true covariance matrix $\Sigma$ compared to the sample covariance matrix, -plays a dual role in this framework. Statistically, it -reduces estimation error by shrinking sample eigenvalues toward their mean. Numerically, the same modification -compresses the eigenvalue spread of the system matrix and directly accelerates CG convergence. For example, on S\&P~500 equity data, using LW shrinkage instead of the sample covariance matrix reduces the number of CG iterations from 744 to 82. -Statistical regularization and spectral preconditioning are, in this -problem, the same operation applied to the same matrix---a connection that does not appear to have been made explicit in -the existing literature. +Second, we show that Ledoit-Wolf shrinkage~\cite{ledoit2004}, proposed to yield a more +accurate estimator of the true covariance matrix $\Sigma$ than the sample covariance +matrix, plays a dual role in this framework. Statistically, it reduces estimation error +by shrinking sample eigenvalues toward their mean. Numerically, the same modification +compresses the eigenvalue spread of the system matrix and directly accelerates CG +convergence (Proposition~\ref{prop:shrinkage}). On S\&P~500 equity data, LW shrinkage reduces the number of CG iterations +from 744 to 82. Statistical regularization and spectral preconditioning are, in this +problem, the same operation applied to the same matrix---a connection that does not +appear to have been made explicit in the existing literature. + +CG of Hestenes and Stiefel~\cite{hestenes1952} is a cornerstone of large-scale +scientific computing; see, e.g., Golub and Van Loan~\cite{golub2013}. To the best of +our knowledge it has not been applied to portfolio optimisation in the matrix-free form +described here. The closest related work applies GMRES to the minimum variance problem +when $\hat \Sigma$ is singular~\cite{bajeux2012}; that formulation still assembles +$X^\top X$ and does not exploit the SPD structure or consider preconditioning. +General-purpose solvers such as Clarabel~\cite{clarabel2024} and OSQP~\cite{osqp} handle +the long-only constraint natively but require an assembled covariance matrix as input. +Markowitz's Critical Line Algorithm~\cite{markowitz1956} traces the full efficient +frontier via simplex-like pivots at $O(n^2)$ each; it targets the entire frontier +rather than a single minimum-variance solve and does not exploit the matrix-free +structure of $\hat\Sigma = X^\top X/T$. + +For completeness, Section~\ref{sec:primaldual} also describes a simplex-projection +strategy based on proximal gradient descent. This method is \emph{not} a Krylov method: +it applies the gradient operator and projects onto the probability simplex at each step +without building a Krylov subspace, and consequently converges in $O(\kappa)$ rather than +$O(\sqrt\kappa)$ iterations. It is included as a matrix-free first-order comparator that +requires no outer active-set loop. -CG of Hestenes and Stiefel \cite{hestenes1952} is a cornerstone of large-scale scientific computing; -e.g., see Golub and van Loan \cite{golub2013}. However, to the best of our knowledge it has not been -applied yet to portfolio optimisation in the matrix-free form described here. -The closest related work applies GMRES to the minimum -variance problem when $\hat \Sigma$ is singular~\cite{bajeux2012}; that -formulation still assembles $X^\top X$ and does not -exploit the SPD structure or consider preconditioning. General-purpose solvers such as -Clarabel~\cite{clarabel2024} handle the long-only constraint natively -but require as input an estimate of the covariance matrix. The remainder of the paper is organized as follows. -\mbox{Sections~\ref{sec:problem}--\ref{sec:solvers}} develop the formulation, shrinkage, and solvers; Section~\ref{sec:results} reports -empirical results on both synthetic and S\&P~500 data. +Sections~\ref{sec:problem}--\ref{sec:solvers} develop the problem formulation, +non-negativity strategies, Ledoit-Wolf preconditioning, and concrete solver +implementations. Section~\ref{sec:results} reports empirical results on S\&P~500 +and synthetic data, benchmarking against direct KKT factorization, Clarabel, OSQP, +and proximal gradient. diff --git a/paper/s2s3_problem.tex b/paper/s2s3_problem.tex index 8eb8b2c..bb9c370 100644 --- a/paper/s2s3_problem.tex +++ b/paper/s2s3_problem.tex @@ -99,14 +99,14 @@ \section{Positive-Definite Reduction}\label{sec:pd} \mathcal{L}(w, \lambda) = \frac{\|Xw\|^2}{T} - \lambda(\mathbf{1}^\top w - 1)~ + \rho \hat \mu~. \] Setting the derivative to zero gives the stationarity condition -$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\hat \mu$, where $\hat\Sigma = X^\top X/T$ (or $\hat \Sigma_{\mathrm{LW}}$ with shrinkage). +$\frac{2}{T}\hat\Sigma\, w = \lambda \mathbf{1} + \rho\hat \mu$, where $\hat\Sigma = X^\top X/T$. For $\rho = 0$, this reduces to $\frac{2}{T}\hat\Sigma\, w = \lambda\mathbf{1}$, determining the solution up to a scalar. \paragraph{Positive-definite reduction} When $\rho = 0$ (pure long-only minimum variance), stationarity gives $2\hat\Sigma\, w = \lambda\mathbf{1}$, so $w$ is a scalar multiple of $\hat\Sigma^{-1}\mathbf{1}$. Setting $v_1 = \hat\Sigma^{-1}\mathbf{1}$ and writing $w = c\,v_1$, the budget constraint -$\mathbf{1}^\top w = 1$ \tcb{scales} +$\mathbf{1}^\top w = 1$ determines $c$: \[ c = \frac{1}{\mathbf{1}^\top v_1} \quad \mbox{and} \quad w = \frac{v_1}{\mathbf{1}^\top v_1}~. \] @@ -121,6 +121,8 @@ \section{Positive-Definite Reduction}\label{sec:pd} In both cases the $(n+1)\times(n+1)$ saddle-point system is never needed and $\lambda$ is recovered analytically rather than as part of the solve. The long-only problem therefore reduces to solving a symmetric positive definite system -$\hat\Sigma_{\mathrm{LW}}\,v = b$ for one or two right-hand sides, subject to the remaining -inequality constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}, -which presents two strategies for doing so. +$\hat\Sigma\,v = b$ for one or two right-hand sides, subject to the remaining inequality +constraint $w \geq 0$; the latter is addressed in Section~\ref{sec:primaldual}, which +presents two strategies for doing so. Section~\ref{sec:precond} shows how Ledoit-Wolf +shrinkage replaces $\hat\Sigma$ with a better-conditioned estimator $\hat\Sigma_{\mathrm{LW}}$ +without changing this reduction. diff --git a/paper/s4_nonneg.tex b/paper/s4_nonneg.tex index ae770c5..5f584f8 100644 --- a/paper/s4_nonneg.tex +++ b/paper/s4_nonneg.tex @@ -1,7 +1,7 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} Section~\ref{sec:pd} reduces the long-only problem to solving an SPD system, leaving -$w \geq 0$ as the sole remaining constraint. Two strategies are available. +$w \geq 0$ as the sole remaining constraint. Two strategies are available. \paragraph{Active-set strategy} The first strategy separates the budget constraint from the non-negativity constraint. @@ -10,7 +10,7 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} solve. In the \emph{primal step}, any asset with a negative weight is dropped from the active -set and the budget-constrained SPD system is re-solved on the reduced universe. Once +set and the budget-constrained SPD system is re-solved on the reduced universe. Once all active weights are non-negative, the \emph{dual step} checks the KKT gradient condition for every excluded asset: global optimality requires every excluded asset $i$ to satisfy @@ -20,9 +20,9 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} \nabla_i f(w) = \tfrac{2}{T}(X^\top Xw)_i - \rho\hat\mu_i\;, \] where $\lambda$ is the budget Lagrange multiplier recovered analytically from -Section~\ref{sec:pd}. An excluded asset that violates this condition would decrease +Section~\ref{sec:pd}. An excluded asset that violates this condition would decrease portfolio variance if held at a small positive weight; it is re-added to the active set -and the loop repeats. The loop terminates when both primal and dual feasibility hold; +and the loop repeats. The loop terminates when both primal and dual feasibility hold; together with stationarity from the inner solve this certifies global optimality. \begin{algorithm}[ht] @@ -67,28 +67,36 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} There is no outer loop: both constraints are handled simultaneously, and the method terminates when the step size $\|w_{k+1} - w_k\|$ falls below a tolerance. -The gradient $\nabla f(w) = \tilde{X}^\top(\tilde{X}w) - \tilde{X}^\top\mathbf{0} -= \tilde{X}^\top(\tilde{X}w)$ is evaluated matrix-free in $O(nT)$ using two -matrix-vector products with $\tilde{X}$, at the same cost per step as the CG inner -solver. The Lipschitz constant $L = \lambda_{\max}(\tilde{X}^\top\tilde{X})$, needed -for the step size $\tau = 1/(2L)$, is estimated once by power iteration on $\tilde{X}$, -again matrix-free. +Although $\hat\Sigma = \tilde{X}^\top\tilde{X}$ admits a stacked-matrix +representation (Section~\ref{sec:precond}), assembling or storing $\tilde{X}$ is +unnecessary. The gradient decomposes additively as +\[ + \nabla f(w) = \hat\Sigma\,w + = \frac{1-\alpha}{T}\,X^\top(Xw) + \alpha\,\hat\Sigma_0\,w\,, +\] +and each term is applied separately: the data contribution costs $O(nT)$ via two +matrix-vector products with $X$, and the target contribution costs $O(n^2)$ for a general +$\hat\Sigma_0$ or $O(n)$ for a scaled identity. The Lipschitz constant +$L = \lambda_{\max}(\hat\Sigma)$, needed for the step size $\tau = 1/(2L)$, +is estimated once by power iteration on the same two-term operator, also matrix-free. \begin{algorithm}[ht] \caption{Simplex-projection solver (proximal gradient, matrix-free)}\label{alg:proximal} \begin{algorithmic}[1] -\Require Stacked matrix $\tilde{X} \in \mathbb{R}^{(T+n_R)\times n}$;\; - tolerance $\varepsilon = 10^{-6}$;\; $k_{\max} = 1{,}000$ +\Require Returns matrix $X \in \mathbb{R}^{T\times n}$;\; shrinkage intensity $\alpha$;\; + target $\hat\Sigma_0 \in \mathbb{R}^{n\times n}$;\; tolerance $\varepsilon = 10^{-6}$ \Ensure Weight vector $w \in \Delta_n$ -\State Estimate $L \leftarrow \lambda_{\max}(\tilde{X}^\top\tilde{X})$ by power iteration - \hfill\textit{(matrix-free, $O(n_{\mathrm{pow}}\cdot nT)$)} +\State Define operator $\mathcal{A}$:\; $v \mapsto \tfrac{1-\alpha}{T}X^\top(Xv) + \alpha\,\hat\Sigma_0 v$ + \hfill\textit{($\hat\Sigma$ applied matrix-free)} +\State Estimate $L \leftarrow \lambda_{\max}(\mathcal{A})$ by power iteration + \hfill\textit{($O(n_{\mathrm{pow}}\cdot nT)$)} \State $\tau \leftarrow (2L)^{-1}$;\; initialise $w_0 \in \mathbb{R}^n$ randomly;\; $k \leftarrow 0$ \Loop - \State $g \leftarrow \tilde{X}^\top(\tilde{X}w_k)$ - \hfill\textit{(matrix-free gradient, $O(nT)$)} + \State $g \leftarrow \mathcal{A}(w_k)$ + \hfill\textit{(gradient, $O(nT)$)} \State $w_{k+1} \leftarrow \Pi_{\Delta_n}(w_k - \tau\,g)$ \hfill\textit{(simplex projection, $O(n\log n)$)} - \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$ \textbf{or} $k \geq k_{\max}$} + \If{$\|w_{k+1} - w_k\|_2 \leq \varepsilon$} \textbf{ break} \EndIf \State $k \leftarrow k + 1$ @@ -101,7 +109,10 @@ \section{Enforcing Non-Negativity}\label{sec:primaldual} The two strategies make different tradeoffs. The active-set loop solves a sequence of linear systems and benefits from fast inner solvers (Section~\ref{sec:solvers}); its convergence to global optimality is certified by the KKT check. The simplex-projection -approach is single-stage and loop-free, but converges in $O(\kappa)$ gradient steps -versus $O(\sqrt{\kappa})$ for CG, so it is most competitive when the condition number is -moderate or the matrix-free $O(nT)$ cost dominates. +approach is single-stage and loop-free; unlike CG, it is not a Krylov method---each +step applies the gradient operator and projects onto $\Delta_n$ without building a Krylov +subspace, so residuals are not orthogonalised against prior directions. This is why its +convergence is $O(\kappa)$ gradient steps versus $O(\sqrt{\kappa})$ for CG; both methods +share the same $O(nT)$ per-step cost, so the $\sqrt{\kappa}$ gap in iteration count +translates directly to a runtime gap on ill-conditioned problems. Section~\ref{sec:solvers} develops concrete implementations of both strategies. diff --git a/paper/solver.tex b/paper/solver.tex index 886c2d0..a320160 100644 --- a/paper/solver.tex +++ b/paper/solver.tex @@ -79,7 +79,7 @@ \subsection{MINRES} The KKT matrix is symmetric but indefinite: the off-diagonal blocks $[A_\mathcal{A}^\top,\, E^\top]$ introduce negative eigenvalues, so the standard Conjugate Gradient (CG) method, which requires symmetric positive definiteness, cannot -be applied directly to the full system. MINRES handles symmetric indefinite systems via +be applied directly to the full system. MINRES handles symmetric indefinite systems via short three-term recurrences and requires only matrix-vector products with the KKT operator: \[ @@ -95,7 +95,7 @@ \subsection{MINRES} \subsection{Conjugate Gradient in the Null Space} An alternative that recovers a symmetric positive definite (SPD) system is to eliminate -the equality constraints explicitly and apply CG to the resulting reduced problem. We +the equality constraints explicitly and apply CG to the resulting reduced problem. We describe the procedure in detail. \paragraph{Setup} @@ -107,7 +107,7 @@ \subsection{Conjugate Gradient in the Null Space} f = \begin{bmatrix} b_\mathcal{A} \\ d \end{bmatrix} \in \mathbb{R}^{s}, \end{equation} where $s = |\mathcal{A}| + \operatorname{rows}(E)$ is the total number of active -constraints. We assume $B$ has full row rank ($s < n$) and $P \succ 0$. +constraints. We assume $B$ has full row rank ($s < n$) and $P \succ 0$. \paragraph{Null-space basis via QR} Compute the full QR factorisation of $B^\top \in \mathbb{R}^{n \times s}$: @@ -133,7 +133,7 @@ \subsection{Conjugate Gradient in the Null Space} \begin{equation}\label{eq:decomp} x = x_p + Q_2\,\tilde{x}, \qquad \tilde{x} \in \mathbb{R}^{n-s}, \end{equation} -since $B(x_p + Q_2\tilde{x}) = f + B Q_2\tilde{x} = f$. Substituting +since $B(x_p + Q_2\tilde{x}) = f + B Q_2\tilde{x} = f$. Substituting \eqref{eq:decomp} into the quadratic objective and differentiating with respect to the free variable $\tilde{x}$: \[ @@ -149,7 +149,7 @@ \subsection{Conjugate Gradient in the Null Space} \end{equation} The reduced matrix $\hat{P} = Q_2^\top P Q_2$ is SPD: $P \succ 0$ implies $\tilde{x}^\top \hat{P}\tilde{x} = (Q_2\tilde{x})^\top P(Q_2\tilde{x}) > 0$ for all -$\tilde{x} \ne 0$ because $Q_2$ has full column rank. CG is therefore directly +$\tilde{x} \ne 0$ because $Q_2$ has full column rank. CG is therefore directly applicable to \eqref{eq:reduced}. \paragraph{Matrix-free reduced operator} @@ -164,18 +164,18 @@ \subsection{Conjugate Gradient in the Null Space} \paragraph{QR setup cost} Computing the full orthogonal factor $Q$ by applying $s$ Householder reflectors to the -$n \times n$ identity costs $O(n^2 s)$ per active-set step. When many inequality +$n \times n$ identity costs $O(n^2 s)$ per active-set step. When many inequality constraints are active---e.g.\ long-only with $s = O(n)$---this reaches $O(n^3)$ and dominates the per-iteration cost, making constraint-eliminated CG slower than MINRES in -that regime. MINRES, by contrast, incurs no such setup cost and has per-iteration cost +that regime. MINRES, by contrast, incurs no such setup cost and has per-iteration cost $O(Tn + sn) = O(Tn)$ for $s \ll T$. \paragraph{Special case: single budget constraint} When $B = \mathbf{1}^\top$ ($s = 1$), the null-space basis $Q_2$ is the Householder -complement of $\hat{e}_1 = \mathbf{1}/\|\mathbf{1}\|$. The maps $Q_2\,\tilde{v}$ and +complement of $\hat{e}_1 = \mathbf{1}/\|\mathbf{1}\|$. The maps $Q_2\,\tilde{v}$ and $Q_2^\top w$ can be applied implicitly via the rank-one reflector $I - 2\hat{e}_1\hat{e}_1^\top$ in $O(n)$, reducing the per-iteration cost from -$O(n^2)$ to $O(Tn)$ and the QR setup to $O(n)$. This is the approach used in the +$O(n^2)$ to $O(Tn)$ and the QR setup to $O(n)$. This is the approach used in the minimum-variance companion paper for the single budget constraint. \paragraph{Recovering the primal solution} diff --git a/src/fast_minimum_variance/__init__.py b/src/fast_minimum_variance/__init__.py index 1282c30..4666502 100644 --- a/src/fast_minimum_variance/__init__.py +++ b/src/fast_minimum_variance/__init__.py @@ -64,4 +64,6 @@ def Problem( # noqa: N802 return _Problem(X, target=target, A=A, b=b, C=C, d=d, alpha=alpha, rho=rho, mu=mu) -__all__ = ["Problem"] +from .data import simulate_equity_returns + +__all__ = ["Problem", "simulate_equity_returns"] diff --git a/src/fast_minimum_variance/_base.py b/src/fast_minimum_variance/_base.py index b39c9b6..df4d6aa 100644 --- a/src/fast_minimum_variance/_base.py +++ b/src/fast_minimum_variance/_base.py @@ -131,8 +131,8 @@ def solve_kkt(self, *, project: bool = True): w = self._clip_and_renormalize(w) return w, iters - def solve_cvxpy(self, *, project: bool = True): - """Solve via CVXPY / Clarabel (reference interior-point solver). + def solve_cvxpy(self, *, project: bool = True, backend: str = "clarabel"): + """Solve via CVXPY with a configurable backend solver. Requires the ``convex`` extra:: @@ -140,9 +140,11 @@ def solve_cvxpy(self, *, project: bool = True): Args: project: Clip and renormalize after solving (see ``solve_kkt``). + backend: CVXPY solver name (default ``"clarabel"``; ``"osqp"`` is + also supported). Returns: - ``(w, n_iters)`` — weight vector of shape ``(N,)`` and Clarabel + ``(w, n_iters)`` — weight vector of shape ``(N,)`` and solver iteration count. Examples: @@ -167,8 +169,9 @@ def solve_cvxpy(self, *, project: bool = True): if self.rho != 0.0 and self.mu is not None: objective = objective - self.rho * (self.mu @ w) + cvxpy_solver = cp.CLARABEL if backend.lower() == "clarabel" else cp.OSQP problem = cp.Problem(cp.Minimize(objective), self._cvxpy_constraints(w, cp)) - problem.solve(solver=cp.CLARABEL) + problem.solve(solver=cvxpy_solver) result = w.value if result is None: @@ -235,6 +238,60 @@ def solve_nnls(self, *, project: bool = True): w = self._clip_and_renormalize(w) return w, iters + def solve_osqp(self, *, project: bool = True): + """Solve via OSQP (operator-splitting QP solver). + + Assembles ``P = 2·Σ_LW`` as a sparse upper-triangular CSC matrix and + calls OSQP directly. Returns ``(w, iters)`` where ``iters`` is the + OSQP iteration count. + + Args: + project: Clip and renormalize after solving (see ``solve_kkt``). + + Returns: + ``(w, n_iters)`` — weight vector of shape ``(N,)`` and number of + OSQP iterations. + + Examples: + >>> import numpy as np + >>> from fast_minimum_variance import Problem + >>> X = np.random.default_rng(0).standard_normal((100, 5)) + >>> w, iters = Problem(X).solve_osqp() + >>> float(round(w.sum(), 6)) + 1.0 + >>> bool((w >= -1e-6).all()) + True + """ + n = self.n + + if self.target is None: + p_dense = 2.0 * ((self.X.T @ self.X) / self.t) + else: + p_dense = 2.0 * ((1 - self.alpha) * (self.X.T @ self.X) / self.t + self.alpha * self.target) + + p_upper = triu(csc_matrix(p_dense), format="csc") + + q = np.zeros(n) + if self.rho != 0.0 and self.mu is not None: + q = -self.rho * self.mu + + a_mat, l_vec, u_vec = self._osqp_constraints() + + solver = osqp.OSQP() + solver.setup( + p_upper, q, a_mat, l_vec, u_vec, + warm_starting=True, + verbose=False, + eps_abs=1e-8, + eps_rel=1e-8, + ) + result = solver.solve() + + w = np.array(result.x) + if project: + w = self._clip_and_renormalize(w) + return w, result.info.iter + def solve_clarabel(self, *, project: bool = True): """Solve via Clarabel interior-point solver (direct API, no CVXPY overhead). @@ -287,17 +344,21 @@ def solve_clarabel(self, *, project: bool = True): def solve_proximal(self, *, project: bool = True): """Solve via proximal gradient descent projected onto the probability simplex. - Minimises ``0.5 * w^T Σ_LW w`` subject to ``w >= 0, sum(w) = 1`` by - casting the objective as a least-squares problem and applying the - Duchi et al. (2008) simplex-projection step. Return tilt (``rho != 0``) - is not supported. + Minimises ``0.5 * w^T Σ_LW w`` subject to ``w >= 0, sum(w) = 1``. + The gradient is computed in two separate terms so that the per-step + cost remains ``O(nT)`` regardless of whether shrinkage is applied: + + grad = (1-α)/T · X^T(Xw) + α · target @ w + + This avoids stacking a (T+n)×n matrix, which would inflate per-step + cost by O(n) under shrinkage. Return tilt (``rho != 0``) is not supported. Args: project: Clip and renormalize after solving (see ``solve_kkt``). Returns: - ``(w, 1)`` — weight vector of shape ``(N,)`` and iteration count - (always 1; the inner iteration count is not exposed). + ``(w, n_iters)`` — weight vector of shape ``(N,)`` and the number + of gradient steps taken. Examples: >>> import numpy as np @@ -311,19 +372,18 @@ def solve_proximal(self, *, project: bool = True): """ from .proximal import prox_gradient - if self.target is None: - mat = self.X / np.sqrt(self.t) - vec = np.zeros(self.t) + if self.target is not None and self.alpha > 0.0: + c = 1.0 - self.alpha + mat = np.sqrt(c) / np.sqrt(self.t) * self.X + alpha, target = self.alpha, self.target + extra_grad = lambda v, a=alpha, tgt=target: a * (tgt @ v) else: - chol_t = np.linalg.cholesky(self.target) - mat = np.vstack([ - np.sqrt(1.0 - self.alpha) / np.sqrt(self.t) * self.X, - np.sqrt(self.alpha) * chol_t.T, - ]) - vec = np.zeros(mat.shape[0]) - - w = prox_gradient(mat, vec) + mat = self.X / np.sqrt(self.t) + extra_grad = None + + vec = np.zeros(self.t) + w, n_iters = prox_gradient(mat, vec, extra_grad=extra_grad) if project: w = self._clip_and_renormalize(w) - return w, 1 + return w, n_iters diff --git a/src/fast_minimum_variance/data/__init__.py b/src/fast_minimum_variance/data/__init__.py new file mode 100644 index 0000000..12f7fb2 --- /dev/null +++ b/src/fast_minimum_variance/data/__init__.py @@ -0,0 +1,5 @@ +"""Synthetic equity return data for benchmarking and testing.""" + +from ._simulate import simulate_equity_returns + +__all__ = ["simulate_equity_returns"] diff --git a/src/fast_minimum_variance/data/_simulate.py b/src/fast_minimum_variance/data/_simulate.py new file mode 100644 index 0000000..4519e4d --- /dev/null +++ b/src/fast_minimum_variance/data/_simulate.py @@ -0,0 +1,79 @@ +"""Simulate equity returns from a latent factor model.""" + +import numpy as np + +__all__ = ["simulate_equity_returns"] + + +def simulate_equity_returns( + n: int, + T: int, + *, + k: int | None = None, + rng: np.random.Generator | int | None = None, +) -> np.ndarray: + """Simulate a T×n demeaned equity return matrix with latent factor structure. + + Returns are generated from the model + + X = F @ B.T + E + + where F (T×k) are factor returns, B (n×k) are factor loadings, and E (T×n) + is idiosyncratic noise. The first factor is a market factor with universally + positive loadings and high variance; the remaining k-1 factors are + style/industry factors with sparse loadings. This produces a covariance + spectrum qualitatively similar to equity universes: a dominant market + eigenvalue, a handful of secondary factor eigenvalues, and a long tail of + near-equal idiosyncratic eigenvalues. + + Parameters + ---------- + n: + Number of assets. + T: + Number of time periods (trading days). + k: + Number of latent factors. Defaults to ``max(3, n // 10)``. + rng: + Random state — a :class:`numpy.random.Generator`, an integer seed, + or ``None`` (non-reproducible). + + Returns + ------- + X : ndarray of shape (T, n) + Demeaned return matrix. Each column has zero mean. + + Examples + -------- + >>> X = simulate_equity_returns(100, 200, k=5, rng=0) + >>> X.shape + (200, 100) + >>> abs(X.mean(axis=0)).max() < 1e-14 + True + """ + rng = np.random.default_rng(rng) + if k is None: + k = max(3, n // 10) + + # Factor volatilities (daily): market ~1 %, style factors ~0.5 % + factor_vols = np.concatenate([[0.01], np.full(k - 1, 0.005)]) + + # Factor returns: T×k + F = rng.standard_normal((T, k)) * factor_vols + + # Factor loadings: n×k + # Market: all assets have positive exposure in [0.4, 0.8] + # Style: sparse (~50 % non-zero), drawn from N(0, 0.2) + B = np.zeros((n, k)) + B[:, 0] = rng.uniform(0.4, 0.8, size=n) + for j in range(1, k): + mask = rng.random(n) < 0.5 + B[mask, j] = rng.standard_normal(int(mask.sum())) * 0.2 + + # Idiosyncratic volatility: uniform in [0.5 %, 1.5 %] per asset + idio_vols = rng.uniform(0.005, 0.015, size=n) + E = rng.standard_normal((T, n)) * idio_vols + + X = F @ B.T + E + X -= X.mean(axis=0) + return X diff --git a/src/fast_minimum_variance/proximal.py b/src/fast_minimum_variance/proximal.py index 9910ce9..0838930 100644 --- a/src/fast_minimum_variance/proximal.py +++ b/src/fast_minimum_variance/proximal.py @@ -78,11 +78,13 @@ def proj_simplex( def _lipschitz( mat: NDArray[np.floating], + extra_matvec=None, n_iter: int = 30, rng: np.random.Generator | None = None, ) -> float: - """Estimate lambda_max(mat.T @ mat) via power iteration (matrix-free). + """Estimate lambda_max(mat.T @ mat + extra) via power iteration (matrix-free). + extra_matvec: optional callable v -> extra @ v for a second SPD contribution. Each iteration costs O(rows * cols) — two matrix-vector products with mat — and never forms the cols x cols normal matrix. """ @@ -93,6 +95,8 @@ def _lipschitz( lip = 1.0 for _ in range(n_iter): w = mat.T @ (mat @ v) + if extra_matvec is not None: + w = w + extra_matvec(v) lip = float(np.linalg.norm(w)) if lip < 1e-15: return lip @@ -103,51 +107,56 @@ def _lipschitz( def prox_gradient( mat: NDArray[np.floating], vec: NDArray[np.floating], + *, + extra_grad=None, eps_rel: float = 1e-6, - max_iter: int = 1000, -) -> NDArray[np.floating]: + max_iter: int = 100000, +) -> tuple[NDArray[np.floating], int]: """Perform proximal gradient descent to solve a constrained optimization problem. Solves the optimization problem: - minimize 0.5 ||mat @ x - vec||^2 + minimize 0.5 ||mat @ x - vec||^2 + g(x) subject to x >= 0, sum(x) = 1 - The gradient mat.T @ (mat @ x) - mat.T @ vec is evaluated matrix-free at each - step; the normal matrix mat.T @ mat is never formed. The Lipschitz constant - L = lambda_max(mat.T @ mat) is estimated once via power iteration, also - matrix-free, at O(n_power_iter * rows * cols) setup cost. + where g captures an optional extra gradient term supplied via ``extra_grad``. + The gradient is evaluated matrix-free at each step; the normal matrix + mat.T @ mat is never formed. The Lipschitz constant is estimated once via + power iteration at O(n_power_iter * rows * cols) setup cost. Parameters ---------- mat : NDArray[np.floating] - A matrix of shape (n_samples, n_features) used in the optimization - problem. + A matrix of shape (n_samples, n_features). vec : NDArray[np.floating] - A vector of shape (n_samples,) used in the optimization problem. + A vector of shape (n_samples,). + extra_grad : callable, optional + v -> additional gradient term (e.g. ``alpha * target @ v`` for + Ledoit-Wolf shrinkage). Must be SPD for convergence guarantees. + When provided, the Lipschitz estimate accounts for this term. eps_rel : float, optional - The relative error threshold for stopping criteria. Default is 1e-6. + Relative step-size change stopping tolerance. Default is 1e-6. max_iter : int, optional - The maximum number of iterations for the algorithm. Default is 1000. + Maximum number of iterations. Default is 100000. Returns: ------- - NDArray[np.floating] - The solution vector of shape (n_features,) obtained after the - optimization process. + tuple[NDArray[np.floating], int] + ``(w, n_iters)`` — weight vector of shape (n_features,) and the + number of gradient steps taken. Examples: -------- >>> import numpy as np >>> mat = np.array([[1.0, 0.5], [0.5, 1.0]]) >>> vec = np.ones(2) - >>> result = prox_gradient(mat, vec) + >>> result, _ = prox_gradient(mat, vec) >>> bool(np.isclose(result.sum(), 1.0)) True """ rng = np.random.default_rng() prim_var: NDArray[np.floating] = np.asarray(rng.standard_normal(size=mat.shape[1])) - lip = _lipschitz(mat, rng=rng) + lip = _lipschitz(mat, extra_matvec=extra_grad, rng=rng) step = 0.5 / lip if lip > 1e-15 else 1.0 # Precompute mat.T @ vec once; zero for minimum-variance (vec = 0). @@ -156,8 +165,10 @@ def prox_gradient( err_rel = eps_rel + 1 while err_rel > eps_rel and ite < max_iter: grad = mat.T @ (mat @ prim_var) - out_prod + if extra_grad is not None: + grad = grad + extra_grad(prim_var) prim_var_new = proj_simplex(prim_var - step * grad) err_rel = float(np.linalg.norm(prim_var - prim_var_new, 2)) prim_var = prim_var_new.copy() ite += 1 - return prim_var + return prim_var, ite diff --git a/tests/test_proximal.py b/tests/test_proximal.py index 7413ad5..cd62955 100644 --- a/tests/test_proximal.py +++ b/tests/test_proximal.py @@ -121,7 +121,7 @@ class TestProxGradient: def test_cla(self, resource_dir) -> None: """Test against CLA expected values.""" covar = np.genfromtxt(resource_dir / "CLA_Data.csv", delimiter=",", skip_header=1)[3:] - result = prox_gradient(covar, np.ones(10)) + result, _ = prox_gradient(covar, np.ones(10)) expected = np.array([0, 0.41200, 0, 0, 0, 0, 0.27612, 0.31188, 0, 0]) assert np.linalg.norm(result - expected, 1) < 1e-5 @@ -130,7 +130,7 @@ def test_output_on_simplex(self) -> None: rng = np.random.default_rng(42) mat = rng.standard_normal((5, 3)) vec = rng.standard_normal(5) - result = prox_gradient(mat, vec) + result, _ = prox_gradient(mat, vec) np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-6) assert np.all(result >= -1e-10) @@ -142,7 +142,7 @@ def test_various_shapes(self, shape: tuple[int, int]) -> None: m, n = shape mat = rng.standard_normal((m, n)) vec = rng.standard_normal(m) - result = prox_gradient(mat, vec) + result, _ = prox_gradient(mat, vec) assert result.shape == (n,) np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) @@ -153,7 +153,7 @@ def test_identity_matrix(self) -> None: n = 5 mat = np.eye(n) vec = np.array([0.1, 0.2, 0.3, 0.2, 0.2]) # Already on simplex - result = prox_gradient(mat, vec) + result, _ = prox_gradient(mat, vec) np.testing.assert_allclose(result, vec, rtol=1e-5) def test_convergence_tolerance(self) -> None: @@ -162,8 +162,8 @@ def test_convergence_tolerance(self) -> None: mat = rng.standard_normal((10, 5)) vec = rng.standard_normal(10) - result_loose = prox_gradient(mat, vec, eps_rel=1e-3) - result_tight = prox_gradient(mat, vec, eps_rel=1e-10) + result_loose, _ = prox_gradient(mat, vec, eps_rel=1e-3) + result_tight, _ = prox_gradient(mat, vec, eps_rel=1e-10) # Both should satisfy constraints np.testing.assert_allclose(result_loose.sum(), 1.0, rtol=1e-3) @@ -175,9 +175,9 @@ def test_max_iter_respected(self) -> None: mat = rng.standard_normal((100, 50)) vec = rng.standard_normal(100) - # Should still produce valid output even with few iterations - result = prox_gradient(mat, vec, max_iter=10) + result, iters = prox_gradient(mat, vec, max_iter=10) assert result.shape == (50,) + assert iters <= 10 def test_objective_convergence(self) -> None: """Test that algorithm converges to similar objective values.""" @@ -189,7 +189,7 @@ def test_objective_convergence(self) -> None: # but objective values should be similar objectives = [] for _ in range(5): - result = prox_gradient(mat, vec, eps_rel=1e-10) + result, _ = prox_gradient(mat, vec, eps_rel=1e-10) obj = 0.5 * np.linalg.norm(mat @ result - vec) ** 2 objectives.append(obj) @@ -200,7 +200,7 @@ def test_near_zero_matrix(self) -> None: """Test with near-zero matrix (low Lipschitz constant).""" mat = np.array([[1e-10, 0], [0, 1e-10]]) vec = np.array([1.0, 1.0]) - result = prox_gradient(mat, vec) + result, _ = prox_gradient(mat, vec) assert result.shape == (2,) np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) @@ -208,7 +208,7 @@ def test_rank_deficient(self) -> None: """Test with rank-deficient matrix.""" mat = np.array([[1, 2, 3], [2, 4, 6]]) # Rank 1 vec = np.array([1.0, 2.0]) - result = prox_gradient(mat, vec) + result, _ = prox_gradient(mat, vec) assert result.shape == (3,) np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) @@ -225,7 +225,7 @@ def test_ill_conditioned(self) -> None: mat = u @ np.diag(s) @ vt vec = np.array([1.0, 1.0, 1.0]) - result = prox_gradient(mat, vec, eps_rel=1e-6, max_iter=5000) + result, _ = prox_gradient(mat, vec, eps_rel=1e-6, max_iter=5000) assert result.shape == (2,) # May not converge perfectly but should satisfy constraints approximately assert abs(result.sum() - 1.0) < 0.01 @@ -238,7 +238,7 @@ def test_random_problems(self, seed: int) -> None: mat = rng.standard_normal((m, n)) vec = rng.standard_normal(m) - result = prox_gradient(mat, vec) + result, _ = prox_gradient(mat, vec) np.testing.assert_allclose(result.sum(), 1.0, rtol=1e-5) assert np.all(result >= -1e-10) @@ -290,7 +290,7 @@ def test_return_shape(self, X) -> None: # noqa: N803 """solve_proximal returns (w, iters) with w of shape (N,).""" w, iters = Problem(X).solve_proximal() assert w.shape == (X.shape[1],) - assert iters == 1 + assert iters >= 1 def test_weights_sum_to_one(self, X) -> None: # noqa: N803 """Weights must sum to 1.""" From 031e586d760568a4ea35550ad3217efdcd1ae9e4 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Mon, 8 Jun 2026 09:10:46 +0400 Subject: [PATCH 25/39] bib file for paper --- paper/bib/refs.bib | 318 +++++++++++++++++++++++++++------------------ 1 file changed, 194 insertions(+), 124 deletions(-) diff --git a/paper/bib/refs.bib b/paper/bib/refs.bib index c4483b0..4173956 100644 --- a/paper/bib/refs.bib +++ b/paper/bib/refs.bib @@ -1,26 +1,45 @@ -@book{nesterov2004, - author = {Nesterov, Yurii}, - title = {Introductory Lectures on Convex Optimization: {A} Basic Course}, - publisher = {Kluwer Academic Publishers}, - address = {Boston, MA}, - year = {2004}, +@article{beck2009, + author = {Beck, Amir and Teboulle, Marc}, + title = {A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems}, + journal = {SIAM Journal on Imaging Sciences}, + year = {2009}, + volume = {2}, + number = {1}, + pages = {183--202}, } -@incollection{simoncini2013, - author = {Simoncini, Valeria and Szyld, Daniel B.}, - title = {On the Superlinear Convergence of {MINRES}}, - booktitle = {Numerical Mathematics and Advanced Applications 2011}, - publisher = {Springer}, - year = {2013}, - pages = { }, - note = {Proceedings of {ENUMATH} 2011}, +@article{bajeux2012, + author = {Bajeux-Besnainou, Isabelle and Bandara, Wimal and Bura, Efstathia}, + title = {A {K}rylov subspace approach to large portfolio optimization}, + journal = {Journal of Economic Dynamics and Control}, + year = {2012}, + volume = {36}, + number = {11}, + pages = {1688--1699}, } -@book{fischer1996, - author = {Fischer, Bernd}, - title = {Polynomial Based Iteration Methods for Symmetric Linear Systems}, - publisher = {Wiley-Teubner}, - year = {1996}, +@article{benzi2005, + author = {Benzi, Michele and Golub, Gene H. and Liesen, Jörg}, + title = {Numerical solution of saddle point problems}, + journal = {Acta Numerica}, + year = {2005}, + volume = {14}, + pages = {1--137}, +} + +@book{bouchaud2003, + author = {Bouchaud, Jean-Philippe and Potters, Marc}, + title = {Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management}, + publisher = {Cambridge University Press}, + edition = {2nd}, + year = {2003}, +} + +@book{boyd2004, + author = {Boyd, Stephen and Vandenberghe, Lieven}, + title = {Convex Optimization}, + publisher = {Cambridge University Press}, + year = {2004}, } @article{bunch1977, @@ -33,30 +52,22 @@ @article{bunch1977 pages = {163--179}, } -@book{boyd2004, - author = {Boyd, Stephen and Vandenberghe, Lieven}, - title = {Convex Optimization}, - publisher = {Cambridge University Press}, - year = {2004}, -} - -@article{benzi2005, - author = {Benzi, Michele and Golub, Gene H. and Liesen, Jörg}, - title = {Numerical solution of saddle point problems}, - journal = {Acta Numerica}, - year = {2005}, - volume = {14}, - pages = {1--137}, +@article{chen2010, + author = {Chen, Yilun and Wiesel, Ami and Eldar, Yonina C. and Hero, Alfred O.}, + title = {Shrinkage Algorithms for {MMSE} Covariance Estimation}, + journal = {{IEEE} Transactions on Signal Processing}, + volume = {58}, + number = {10}, + pages = {5016--5029}, + year = {2010}, } -@article{bajeux2012, - author = {Bajeux-Besnainou, Isabelle and Bandara, Wimal and Bura, Efstathia}, - title = {A {K}rylov subspace approach to large portfolio optimization}, - journal = {Journal of Economic Dynamics and Control}, - year = {2012}, - volume = {36}, - number = {11}, - pages = {1688--1699}, +@phdthesis{choi2006, + author = {Choi, Sou-Cheng T.}, + title = {Iterative Methods for Singular Linear Equations and Least-Squares Problems}, + school = {Stanford University}, + year = {2006}, + type = {PhD thesis}, } @article{clarabel2024, @@ -70,6 +81,26 @@ @article{clarabel2024 pages = {481--499}, } +@article{condat2016, + author = {Condat, Laurent}, + title = {Fast projection onto the simplex and the $\ell_1$ ball}, + journal = {Mathematical Programming}, + year = {2016}, + volume = {158}, + number = {1--2}, + pages = {575--585}, +} + +@article{demiguel2009, + author = {DeMiguel, Victor and Garlappi, Lorenzo and Nogales, Francisco J. and Uppal, Raman}, + title = {A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms}, + journal = {Management Science}, + year = {2009}, + volume = {55}, + number = {5}, + pages = {798--812}, +} + @article{diamond2016, author = {Diamond, S. and Boyd, S.}, title = {{CVXPY}: A {Python}-embedded modeling language for convex optimization}, @@ -80,6 +111,16 @@ @article{diamond2016 pages = {1--5}, } +@inproceedings{duchi2008, + author = {Duchi, John and Shalev-Shwartz, Shai and Singer, Yoram and Chandra, Tushar}, + title = {Efficient Projections onto the $\ell_1$-Ball for Learning in High Dimensions}, + booktitle = {Proceedings of the 25th International Conference on Machine Learning ({ICML})}, + year = {2008}, + pages = {272--279}, + publisher = {ACM}, + address = {New York, NY}, +} + @misc{fast_minimum_variance, author = {Schmelzer, T.}, title = {fast-minimum-variance: Solving Minimum Variance Portfolios Fast}, @@ -88,6 +129,13 @@ @misc{fast_minimum_variance note = {{MIT} License}, } +@book{fischer1996, + author = {Fischer, Bernd}, + title = {Polynomial Based Iteration Methods for Symmetric Linear Systems}, + publisher = {Wiley-Teubner}, + year = {1996}, +} + @book{golub2013, author = {Golub, Gene H. and Van Loan, Charles F.}, title = {Matrix Computations}, @@ -96,6 +144,54 @@ @book{golub2013 year = {2013}, } +@book{greenbaum1997, + author = {Greenbaum, Anne}, + title = {Iterative Methods for Solving Linear Systems}, + publisher = {SIAM}, + address = {Philadelphia}, + year = {1997}, + series = {Frontiers in Applied Mathematics}, + volume = {17}, +} + +@article{hestenes1952, + author = {Hestenes, Magnus R. and Stiefel, Eduard}, + title = {Methods of conjugate gradients for solving linear systems}, + journal = {Journal of Research of the National Bureau of Standards}, + year = {1952}, + volume = {49}, + number = {6}, + pages = {409--436}, +} + +@article{laloux1999, + author = {Laloux, Laurent and Cizeau, Pierre and Bouchaud, Jean-Philippe and Potters, Marc}, + title = {Noise Dressing of Financial Correlation Matrices}, + journal = {Physical Review Letters}, + year = {1999}, + volume = {83}, + number = {7}, + pages = {1467--1470}, +} + +@book{lawson1974, + author = {Lawson, Charles L. and Hanson, Richard J.}, + title = {Solving Least Squares Problems}, + publisher = {Prentice-Hall}, + year = {1974}, + address = {Englewood Cliffs, NJ}, +} + +@article{ledoit2003jpm, + author = {Ledoit, Olivier and Wolf, Michael}, + title = {Honey, {I} Shrunk the Sample Covariance Matrix}, + journal = {Journal of Portfolio Management}, + year = {2004}, + volume = {30}, + number = {4}, + pages = {110--119}, +} + @article{ledoit2004, author = {Ledoit, O. and Wolf, M.}, title = {A well-conditioned estimator for large-dimensional covariance matrices}, @@ -116,14 +212,24 @@ @article{ledoit2020 pages = {3043--3065}, } -@article{merton1972, - author = {Merton, Robert C.}, - title = {An Analytic Derivation of the Efficient Portfolio Frontier}, - journal = {Journal of Financial and Quantitative Analysis}, - year = {1972}, - volume = {7}, +@article{marchenko1967, + author = {Marchenko, Vladimir A. and Pastur, Leonid A.}, + title = {Distribution of eigenvalues for some sets of random matrices}, + journal = {Mathematics of the USSR-Sbornik}, + year = {1967}, + volume = {1}, number = {4}, - pages = {1851--1872}, + pages = {457--483}, +} + +@article{markowitz1952, + author = {Markowitz, H.}, + title = {Portfolio selection}, + journal = {Journal of Finance}, + year = {1952}, + volume = {7}, + number = {1}, + pages = {77--91}, } @article{markowitz1956, @@ -136,24 +242,41 @@ @article{markowitz1956 pages = {111--133}, } -@article{markowitz1952, - author = {Markowitz, H.}, - title = {Portfolio selection}, - journal = {Journal of Finance}, - year = {1952}, +@article{merton1972, + author = {Merton, Robert C.}, + title = {An Analytic Derivation of the Efficient Portfolio Frontier}, + journal = {Journal of Financial and Quantitative Analysis}, + year = {1972}, volume = {7}, - number = {1}, - pages = {77--91}, + number = {4}, + pages = {1851--1872}, } -@article{hestenes1952, - author = {Hestenes, Magnus R. and Stiefel, Eduard}, - title = {Methods of conjugate gradients for solving linear systems}, - journal = {Journal of Research of the National Bureau of Standards}, - year = {1952}, - volume = {49}, - number = {6}, - pages = {409--436}, +@book{nesterov2004, + author = {Nesterov, Yurii}, + title = {Introductory Lectures on Convex Optimization: {A} Basic Course}, + publisher = {Kluwer Academic Publishers}, + address = {Boston, MA}, + year = {2004}, +} + +@book{nocedal2006, + author = {Nocedal, Jorge and Wright, Stephen J.}, + title = {Numerical Optimization}, + edition = {2nd}, + publisher = {Springer}, + address = {New York}, + year = {2006}, +} + +@article{osqp, + author = {Stellato, Bartolomeo and Banjac, Goran and Goulart, Paul and Bemporad, Alberto and Boyd, Stephen}, + title = {{OSQP}: An Operator Splitting Solver for Quadratic Programs}, + journal = {Mathematical Programming Computation}, + volume = {12}, + number = {4}, + pages = {637--672}, + year = {2020}, } @article{paige1975, @@ -166,31 +289,6 @@ @article{paige1975 pages = {617--629}, } -@book{greenbaum1997, - author = {Greenbaum, Anne}, - title = {Iterative Methods for Solving Linear Systems}, - publisher = {SIAM}, - address = {Philadelphia}, - year = {1997}, - series = {Frontiers in Applied Mathematics}, - volume = {17}, -} - -@phdthesis{choi2006, - author = {Choi, Sou-Cheng T.}, - title = {Iterative Methods for Singular Linear Equations and Least-Squares Problems}, - school = {Stanford University}, - year = {2006}, - type = {PhD thesis}, -} - -@unpublished{minvar_paper, - author = {Schmelzer, Thomas and Wolf, Michael and Stoll, Martin}, - title = {Krylov Methods for the Minimum Variance Portfolio}, - note = {Working paper, Jebel Quant}, - year = {2026}, -} - @phdthesis{schmelzer2004, author = {Schmelzer, Thomas}, title = {Block {K}rylov methods for {H}ermitian linear systems}, @@ -199,40 +297,12 @@ @phdthesis{schmelzer2004 type = {Diploma thesis}, } -@article{condat2016, - author = {Condat, Laurent}, - title = {Fast projection onto the simplex and the $\ell_1$ ball}, - journal = {Mathematical Programming}, - year = {2016}, - volume = {158}, - number = {1--2}, - pages = {575--585}, -} - -@inproceedings{duchi2008, - author = {Duchi, John and Shalev-Shwartz, Shai and Singer, Yoram and Chandra, Tushar}, - title = {Efficient Projections onto the $\ell_1$-Ball for Learning in High Dimensions}, - booktitle = {Proceedings of the 25th International Conference on Machine Learning ({ICML})}, - year = {2008}, - pages = {272--279}, - publisher = {ACM}, - address = {New York, NY}, -} - -@book{lawson1974, - author = {Lawson, Charles L. and Hanson, Richard J.}, - title = {Solving Least Squares Problems}, - publisher = {Prentice-Hall}, - year = {1974}, - address = {Englewood Cliffs, NJ}, -} - -@article{osqp, - author = {Stellato, Bartolomeo and Banjac, Goran and Goulart, Paul and Bemporad, Alberto and Boyd, Stephen}, - title = {{OSQP}: An Operator Splitting Solver for Quadratic Programs}, - journal = {Mathematical Programming Computation}, - volume = {12}, - number = {4}, - pages = {637--672}, - year = {2020}, +@incollection{simoncini2013, + author = {Simoncini, Valeria and Szyld, Daniel B.}, + title = {On the Superlinear Convergence of {MINRES}}, + booktitle = {Numerical Mathematics and Advanced Applications 2011}, + publisher = {Springer}, + year = {2013}, + pages = { }, + note = {Proceedings of {ENUMATH} 2011}, } From 2e4511135c7965ae7d54a7e136ff88f9b203fc34 Mon Sep 17 00:00:00 2001 From: Thomas Schmelzer Date: Mon, 8 Jun 2026 09:13:29 +0400 Subject: [PATCH 26/39] further cleaning --- 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