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 $i,j \in 1, 2,\dots,k$?
1 Answer
Too long to comment. See if this helps.
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.

$\begingroup$ Yes, this is basically the standard formulation for the spectral norm (squared) of an affine function of the optimization variable. So using a convex optimization tool such as CVXPY, you could just use the builtin spectral norm function applied to A+BXC, and add the other constraints. Then square the optimal objective value. $\endgroup$ Commented Apr 18, 2022 at 1:02