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 $i,j \in 1, 2,\dots,k$?
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$\begingroup$ Without restrictions on $X$? $\endgroup$– A rural readerCommented Feb 28, 2021 at 0:56
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$\begingroup$ @FedericoPoloni yes for all the components. $\endgroup$– hichem hbCommented Feb 28, 2021 at 13:05
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1 Answer
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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.
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$\begingroup$ 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 built-in 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