A matrix norm optimization problem involving matrix inversion

Question

I am trying to characterize the the matrix $X^*$ that solves the following optimization (that is an analytical or semi-analytical solution form).

\begin{equation} X^*=\arg\min_X \| (A+X)^{-1}\|_F \quad\mbox{s.t.}\quad \|X\|_p=r , X \succeq 0 \end{equation}

$A$ and $X$ are both symmetric square matrices. In fact, $A$ is positive definite and $X$ is positive semi-definite. Here $\|.\|_F$ and $\|.\|_p$ are Frobenius and p-Schatten norms.

I am interested in $X^*$ when $p=1$ and $p=2$ (but even knowing the solution in one of these cases is still very helpful).

The problem statement itself looks simple, but I have not been able to characterize its solution. Any thoughts are greatly appreciated.

Thanks!

Golabi

Answer 1

Presuming (as discussed in the question comments), the constraint is $\|X\|_p \le r$ rather than $\|X\|_p=r$, this can be numerically solved in CVX.

Formulation is described in http://ask.cvxr.com/t/represent-schatten-p-norm-in-cvx/649 .