Add interconversion between SDPA and SeDuMi

Author: usamamuneeb

docs/formats/index.md

@@ -68,8 +68,8 @@ The SeDuMi solver solves the following primal ($$\mathcal{P}$$) and dual ($$\mat
 $$
 \begin{aligned}
 (\mathcal{P}) \min_{x}. \quad & c^T x\\
-\textrm{s.t.} \quad & x \succeq_K 0\\
- & A x - b = 0
+\textrm{s.t.} \quad & A x - b = 0\\
+ & x \succeq_K 0
 \end{aligned}
 $$
 
@@ -93,6 +93,8 @@ A commonly used dataset for problems in SeDuMi format is the [DIMACS](http://arc
 
 The primal-dual pair in SeDuMi is the reverse of that in SDPA. In SeDuMi, the dual problem is a linear-matrix-inequality form.
 
+To see the relationship between SDPA and SeDuMi formats, please see the subsection on the [Interconversion between SDPA and SeDuMi Formats](sdpa_sedumi.html).
+
 # CLP Format
 
 SDPAP (of which this wrapper is a fork) introduced a third format, called CLP that stands for Conic-form Linear Optimization Problems. The official specification of this format is given in the [user manual of SDPAP](https://sourceforge.net/projects/sdpa/files/sdpa-p/sdpap_manual.pdf) on the [official SDPA website](http://sdpa.sourceforge.net/download.html).
@@ -102,8 +104,8 @@ The CLP format is defined both as a file format, as well as an in memory represe
 $$
 \begin{aligned}
 (\mathcal{P}) \min_{x}. \quad & c^T x\\
-\textrm{s.t.} \quad & x \succeq_K 0\\
- & A x - b \succeq_J 0
+\textrm{s.t.} \quad & A x - b \succeq_J 0\\
+ & x \succeq_K 0
 \end{aligned}
 $$
 

docs/formats/sdpa_sedumi.md

@@ -0,0 +1,104 @@
+---
+layout: default
+title: SDPA-SeDuMi Interconversion
+parent: SDP Formats
+nav_order: 2
+---
+
+# Interconversion between SDPA and SeDuMi Formats
+
+As discussed on the [SDPA Formats](index.html) page, the SDPA and SeDuMi formats are reverses of each other. 
+In SDPA, the primal problem is a linear-matrix-inequality form while in SeDuMi, the dual problem is a linear-matrix-inequality form.
+
+It's also discussed that SDPA has the ability to solve the problem in either format. To this end, SDPA comes with a preprocessor directive `REVERSE_PRIMAL_DUAL`.
+
+- With `REVERSE_PRIMAL_DUAL = 1`, we use the SDPA form. This is the default setting (i.e. SDPA is configured to treat the primal problem as a linear-matrix-inequality form).
+- With `REVERSE_PRIMAL_DUAL = 0`, we use the SeDuMi form.
+
+Below we show how setting `REVERSE_PRIMAL_DUAL = 0` gives us the SeDuMi format. We start off by writing the primal-dual pair that SDPA solves with the default setting (i.e. with `REVERSE_PRIMAL_DUAL = 1`).
+
+$$
+\begin{aligned}
+(\mathcal{P}) \min_{x}. \quad & c^T x\\
+\textrm{s.t.} \quad & \sum_{i=1}^m F_i x_i - F_0 \succeq_{\mathbb{S}^n_+} 0
+\end{aligned}
+$$
+
+$$
+\begin{aligned}
+(\mathcal{D}) \max_{Y}. \quad & F_0 \cdot Y\\
+\textrm{s.t.} \quad & c_i - F_i \cdot Y = 0, \quad i = 1, ..., m \\
+ & Y \succeq_{\mathbb{S}_+} 0
+\end{aligned}
+$$
+
+The primal variable (i.e. vector $$x$$) with `REVERSE_PRIMAL_DUAL = 1` should become the dual variable with `REVERSE_PRIMAL_DUAL = 0` and the dual variable (i.e. the matrix $$Y$$) should become the primal variable. To this end, we will make the below substitutions.
+
+$$
+\begin{aligned}
+x &\longrightarrow y\\
+Y &\longrightarrow X\\
+\end{aligned}
+$$
+
+To achieve similarity with SeDuMi, we will also perform the below substitutions.
+
+$$
+\begin{aligned}
+F_i &\longrightarrow -A_i\\
+c &\longrightarrow -b\\
+\end{aligned}
+$$
+
+After these substitutions, we get the pair
+
+$$
+\begin{aligned}
+(\mathcal{?}) \min_{y}. \quad & -b^T y\\
+\textrm{s.t.} \quad & \sum_{i=1}^m -A_i y_i + A_0 \succeq_{\mathbb{S}^n_+} 0
+\end{aligned}
+$$
+
+$$
+\begin{aligned}
+(\mathcal{?}) \max_{X}. \quad & -A_0 \cdot X\\
+\textrm{s.t.} \quad & -b_i + A_i \cdot X = 0, \quad i = 1, ..., m \\
+ & X \succeq_{\mathbb{S}_+} 0
+\end{aligned}
+$$
+
+We can now flip the $$\min$$ and $$\max$$ and remove the minus signs from the objectives. Since by convention the primal problem is a minimization problem and the dual is a maximization problem, we also exchange their roles as primal and dual and create the pair corresponding to `REVERSE_PRIMAL_DUAL = 0` as below:
+
+$$
+\begin{aligned}
+(\mathcal{P'}) \min_{X}. \quad & A_0 \cdot X\\
+\textrm{s.t.} \quad & A_i \cdot X = b_i, \quad i = 1, ..., m \\
+ & X \succeq_{\mathbb{S}_+} 0
+\end{aligned}
+$$
+
+$$
+\begin{aligned}
+(\mathcal{D'}) \max_{y}. \quad & b^T y\\
+\textrm{s.t.} \quad & A_0 - \sum_{i=1}^m A_i y_i \succeq_{\mathbb{S}^n_+} 0
+\end{aligned}
+$$
+
+In general, conversion from matrices to vectors and vice versa can be obtained very easily using the $$\text{vec}(...)$$ and $$\text{vec}^{-1}(...)$$ operations. Knowing that $$\text{vec}(X)$$ in $$(\mathcal{P'})$$ above becomes $$x$$ in $$(\mathcal{P''})$$ below, and the fact that SDP cone is self dual (i.e. $$\mathbb{S}^n_+ = {\mathbb{S}^n_+}^*$$) we now see the similarity with [the SeDuMi format](index.html#sedumi-format):
+
+$$
+\begin{aligned}
+(\mathcal{P''}) \min_{x}. \quad & c^T x\\
+\textrm{s.t.} \quad & A x - b = 0\\
+ & x \succeq_K 0
+\end{aligned}
+$$
+
+$$
+\begin{aligned}
+(\mathcal{D''}) \max_{y}. \quad & b^T y\\
+\textrm{s.t.} \quad & c - A^T y \succeq_{K^*} 0
+\end{aligned}
+$$
+
+Another difference is that the cone $$K$$ supported by SeDuMi is a generalization of the SDP cone, in sense that it also supports SOCP constraints.