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.
You can make a block-diagonal matrix with all these matrices as its blocks. The newly made matrix is PDS iff all these blocks are SDP. Of course the size of the matrix is so large, but number of PSD conditions decreases to 1.