# Multiple SDP constraints

I have the following semidefinite program with $$N$$ semidefinite constraints,

\begin{equation*} \begin{aligned} \min_{\theta \in \mathbb{R},\; 0 \leq w \leq 1}&\quad \theta\\ \text{st:}&\quad \left[ \begin{array}{cc} \theta & x^{\top}_i \\ x_i & X^{\top} \text{diag(w)} X \end{array} \right] \succeq 0, \quad i = 1,\ldots, N \end{aligned} \end{equation*}

I want to ask if it is possible to reformulate the constraints to something simpler. A single SDP constraint, or something simpler that an SDP constraint. As $$N$$ grows, and the dimension of $$X$$ grows, the problem becomes very large and slow to solve.

• Can you state precisely the shape of your variables / constants as it is quite ambiguous at the moment? As I read the problem now, the size of the SDP grows with $N$ but I don't see why the dimension of $X$ grows with $N$. Also, if you want to trivially convert a collection of $Y_i \succeq 0$ constraints into a single SDP constraint then you can just convert it to a block diagonal matrix, i.e., $\bigoplus_i Y_i \succeq 0$. This won't reduce the size of your problem though. Commented Sep 23, 2021 at 8:46