I would like to find the symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ which has been proven to be convex in Convexity of $\mathrm{trace}(S) + m^2\mathrm{trace}(S^{-2})$ given the constraint that $S\in \Omega=\{S\in \mathcal{M}_{m,m}, S=(B-AC)^T(B-AC) \quad| \quad\|A\|_1\leq \delta, A\in \mathcal{M}_{n,n} \}$ which is my space of solution set where $B\in \mathcal{M}_{n,m}$ and $C\in \mathcal{M}_{n,m}$ are given matrices and $A$ is unknown but we're given $A^{true}$ to compare $S^{estimated}$ vs. $S^{true}=(B-A^{true}C)^T(B-A^{true}C)$. This is not a convex problem since my solution space $\Omega$ is not a convex set.
we have the explicit expression of $S=B^TB + B^T(AC)-(AC)^TB-B^TAC+(AC)^TAC$ where $\|A\|_1\leq \delta$.
And so I'm trying to reformulate my problem and may be we can disregard some of the solutions by considering a convex set of solutions instead of $\Omega$.
What could be this new convex set that guarantees the convergence with respect to the expression of $S$ and the $L_1$ constraint on $A$.