I have an engineering background. At work, I came across the following problem
\begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ s.t.~&\left(\mathbf{A}_0+\sum_{i=1}^{K}y_i\mathbf{A}_i\right)-\lambda\mathbf{I} \succeq 0 \end{align}
where $\mathbf{A}_i$ are all Hermitian matrices. I know that this is called a Linear Matrix Inequality (LMI) problem and can be solved by a general convex package (e.g., CVX). To me, it seems like we are looking for a matrix formed from the linear combination of given Hermitian matrices whose smallest eigenvalue is as maximum as possible among all such combinations. I was wondering if they are iterative algorithms to solve this problem which are simple to implement. Please point me to relevant references.