Variable equality constraints in SDP Problem

Question

I'm quite new to SDP programming, hence I might not have been able to use the right search terms to find a solution.

I try to reformulate an SDP problem to the original form. However a side constraint gives me some headache. Given the problem, where $X$ is a blockmatrix, which can be written like: $$X = \begin{bmatrix}A_1 & B_1^T&0&0 \\ B_1& C &0&0 \\ 0 &0& A_2 & B_2^T \\ 0&0&B_2 & C\end{bmatrix}$$ with $A_1,B_1,A_2,B_2,C$ all being matrices. The SDP problem can be formulated as follows:

$$\underset{X}{\text{min }} \langle W, X \rangle, \\ X\succeq 0$$ and $W$ is a constant matrix However I am wondering, how I can enforce, that the two $C$ matrices in $X$ are the same after solving? For the problem it is important, that it is formulated like a standard SDP problem.

Answer 1

First, a standard semidefinite program (in primal form) would be

$$\underset{X}{\text{min }} \langle W, X \rangle, \\ X\succeq 0,~\mathbf{A}(X) = b$$

where $\mathbf{A}(X) = b$ denotes the primal equalities (the model you have written is either trivially solved by $X=0$ or is unbounded)

In reality one would never work with such a limited form, so you can at least assume you can work with a cone $X$ composed of a direct product of several cones, in your case 2, call them $X_1$ and $X_2$. What you have to do then is simply to sit down and write down all the equalities describing that certain elements in $X_1$ and $X_2$ are the same, i.e. define all the linear equalities $\mathbf{A}(X_1,X_2) = b$. If you absolutely refuse to use anything but a (too) standard form where you only can have one cone, you not only have to create all the equalities equating two blocks in $X$, but also a lot of equalities describing zero elements.

..and in practice you would be much better off using a modelling language for this and skip all the indexing and book-keeping head-ache.