Maximum determinant of a sum subject to singular value constraint

Question

Let $| \cdot |$ denote the determinant of a matrix.

I am interested if questions of the following type have been addressed:

Let $A$ be some symmetric positive definite matrix. \begin{align} \max_{C: C \text{ is psd, symmetric, and } \sum_i \sigma_i(C) =a } |A+C| \end{align} where $\sigma_i$ is the $i$ singular value of $C$.

At this point, I don't even know what literature to look at in order to solve this problem.

I would be grateful for any suggestion such as how to solve it and what to read next.

Answer 1

Unless you want an analytic solution (would not be surprised if there is one), this is a standard MAXDET problem (maximization of determinant of psd matrix, with psd constraints etc)

Working in MATLAB and the Toolbox YALMIP (disclaimer, developed by me) and an SDP solver available, it would be something like this

n = 5;
A = randn(n);A = A*A';
C = sdpvar(n);
a = 1;
optimize([trace(C) == a, C>=0],-logdet(C+A))