optimization of inverse matrix with constraint on matrix elements

Question

everyone! I have this optimization problem with constraint. $D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter. $x$ and $v$ are two known p-dimensional vectors. The objective function is $D=\mathrm{argmin}\, x^T(D+T)^{-1}x$. Assuming each row vector of $D$ is $D_i=(D_{i1},\ldots,D_{ip})$, the constraints are $|D_{i1}|+|D_{i2}|+...+|D_{ip}|<=|v_i|$. I think the constraint can be written as linear inequality by letting $\beta=\beta_+-\beta_-$ and $|\beta|=\beta_++\beta_-$. The problem is the objective function includes the inverse of $D+T$. It keeps bugging me for a long time. Is there any available algorithm for this kind of problem? Or is this kind of problem unsolvable? I do appreciate your suggestion!

Answer 1

Let $B = xx^T$, and $Y=(D+T)$, then $x^TY^{-1}x=\mbox{Tr}[Y^{-1}B]=\mbox{Tr}[\sqrt{B}^T Y^{-1} \sqrt{B}]$, where $B=\sqrt{B}\sqrt{B}^T$ is the Cholesky factorization. Note that $$\min \mbox{Tr}[\sqrt{B}^T Y^{-1} \sqrt{B}] = \min \{\mbox{Tr}(F):\, F\succeq \sqrt{B} Y^{-1} \sqrt{B} \}. $$ Finally, the latter optimization problem is semidefinite representable. By the Lemma on Schur complements, we can re-write this problem as $$ \min \{\mbox{Tr}(F):\, \left[ \begin{array}{cc} (D+T) & \sqrt{B}\\ \sqrt{B}^T & F \end{array}\right] \succeq 0 \}. $$ Your bound constraints can be easily added to the formulation above.

I hope it helps