# Minimizing the largest eigenvalue of matrix product

Let $$A\in\mathbb{C}^{m\times n}$$, $$B\in\mathbb{C}^{n\times k}$$, $$C\in\mathbb{C}^{k\times m}$$ be given complex matrices. The objective of the optimization problem is $$\begin{equation} \mathop {\arg \min }\limits_X \lambda_{\max} \left( (A + BXC)(A + BXC)^H \right), \end{equation}$$ where $$X\in\mathbb{C}^{k\times k}$$ is a matrix with $$|x(i,j)|<1$$ for all $$i,j \in 1, 2,\dots,k$$?

• Without restrictions on $X$? Feb 28, 2021 at 0:56
• @FedericoPoloni yes for all the components. Feb 28, 2021 at 13:05

Consider the following convex optimization problem: $$\hspace{-2cm}\min~~~~~~~~d\\ \hspace{-2cm}\mbox{subject to}\\ \hspace{6cm}\begin{bmatrix} dI & A + BXC\\ \left(A + BXC\right)^H & I \end{bmatrix} \succeq 0\\ \hspace{6cm}|X(i,j)|\leq 1, \forall i,j.$$ By Schur's complement, we have that the LMI is valid iff $$dI \succeq \left(A + BXC\right) \left(A + BXC\right)^H.$$
For a given $$X$$ the minimum $$d$$ that satisfies the LMI is equal to $$\lambda_{\max}\left(\left(A + BXC\right) \left(A + BXC\right)^H\right)$$. Hence, solving the convex opt problem (using say CVXPY) should give the desired solution.