I have an engineering back ground. Due to work, I came across this 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}\geq 0
\end{align}
where $\mathbf{A}_i$ are all hermitian matrices. We are seeking $\lambda$ and $y_i$. I know that this is called a Linear Matrix Inequality problem and can be solved by a general convex package (for eg, 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.
[Math] Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices
convex optimizationeigenvectorlinear algebramatricesoc.optimization-and-control
Best Answer
Two iterative algorithms that solve LMI problems is the ellipsoid algorithm and interior-point methods. Both are described in sections 2.3 and 2.4 of Stephen Boyd's book "Linear Matrix Inequalities in System and Control Theory" [1] and in the references therein.
See also [2] for already implemented solvers. In particular, if you use MATLAB I recommend using the SeDuMi solver with the YALMIP parser [3], since this one allows one to input the LMI programs to the solver in a more intuitive way.