A matrix optimization problem that resembles a standard semidefinite program

Question

I have a constrained matrix optimization problem as follows

\begin{align} \max\limits_{X} \;\; &\mbox{tr}\Big( \left(C - \frac{1}{2} B R^{-1} S \right) \Lambda X^T \Big) \\ \text{subject to} \;\; &\left[ \begin{array}{ll} -\frac{1}{2}R^{-1}S\Lambda X^T & X \Lambda^{1/2} \\ \Lambda^{1/2} X^T & I \end{array} \right] \succeq 0 \\ &R^{-1}S\Lambda X^T = (R^{-1}S\Lambda X^T)^T \end{align}

where $R$ is symmetric and $\Lambda$ is symmetric and positive-semi-definite.

I am trying to prove that this is a semidefinite program. The objective is linear in the entries of the matrix variable $X$ and I have a PSD constraint. However, I read here that a standard SDP involves minimization of a linear objective function subject to an affine combination of symmetric matrices being positive-semidefinite. I can prove that when $X$ is scalar, this program is indeed an SDP according to this criterion. However, for the general matrix case, I cannot show that the PSD constraint involves an affine combination of symmetric matrices. Moreover, I do not know if the equality constraint can be handled in a standard SDP. I would appreciate any thoughts on this.

Answer 1

You are reading about one standard form, and about one side of the primal-dual pairs. These are not set in stone, although they all are about linearly parameterized matrices constrained to be semidefinite.

One way to define a standard semidefinite program (from the dual, LMI form) is $\max_y b^Ty$ subject to $\mathcal{C}-\mathcal{A}(y)\succeq 0, Fy = f$ where $\mathcal{C}-\mathcal{A}(y)$ represent a set of linear matrix inequalities in $y$. Solvers define problems through the data $\{b,\mathcal{C},\mathcal{A},F,f\}$.

In your case, introduce a symmetric matrix $Z$ to replace $\frac{1}{2}R^{-1}S\Lambda X^T$, add the associated equality, and you have a semidefinite program in required form, with $y$ being the unique elements in $X$ and $Z$.