To optimize a PSD $(n,n)$ matrix $ X $, we are asked us to propose (give ), implement and compare different Algorithms for the resolution of a Linear matrix inequality given by:
$$\begin{array}{ll} \text{minimize} & t\\ \text{subject to} & \left( {\begin{array}{*{20}{c}} X & B\\ {{B^T}}&{C + t I}\end{array}}\right) \succeq 0\\ & X \succeq 0\\ & t \geq 0\end{array}$$
where $C$ is an $m \times m$ symmetric matrix and $B$ is an $n \times m$ matrix, and $I$ is the identity matrix.
The objective is to find the PSD matrix $X$ and the parametre $t$ that satisfy those constraints.