I have a graph with edge-set E and the following SDP (page 8 here)
I'm trying to use CVXOPT to solve this problem, which asks for SDP to be expressed in vectorized form as below:
How do I go about turning my SDP into this form?
Update From suggestion of Joachim Dahl it's easier to figure out values from the dual form. We need to set $G_1,h_1$ and $c$ as follows
$G_1=\left(\begin{array}{c} \text{vec}(I)\\\\ \text{vec}(U_{e1})\\\\ \text{vec}(U_{e2})\\\\ \cdots\\\\ \text{vec}(U_{em}) \end{array}\right)$
here $I$ is $n\times n$ identity matrix, $\text{vec}(A)$ is matrix $A$ taken as vector in row-major vector form, $U_{ek}$ is a $0,1$-valued matrix where entry $i,j$ is 1 iff $i,j$ is in the edge $ek$.
$c = (-1,0,0,0,0,\ldots,0)$
$h_1=- \left(\begin{array}{ccc} 1&1&\ldots\\\\ 1&1&\ldots\\\\ \ldots&\ldots&\ldots \end{array} \right)$
Note that for a graph with $n$ nodes and $m$ edges, $G_1$ is an $m+1\times n^2$ matrix, $c$ is a vector of length $m+1$ and $h_1$ is an $n\times n$ matrix. Matrix $X$ from original formulation is the $z_1$ parameter, an $n^2$ vector representing the matrix in row-major form