# optimization of inverse matrix with constraint on matrix elements

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!

• I think you meant beta=(beta+)-(beta-) and |beta|=(beta+)+(beta-). – Cristóbal Guzmán Apr 5 '14 at 11:13
• Have you tried to formulate the optimality system, e.g. by using this: en.wikipedia.org/wiki/… – Dirk Feb 2 '15 at 8:36

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.
• Why are you introducing $B$ and $F$? Introduce a scalar $t$ and simply minimize $t$ subject to $\begin{bmatrix}D+T & x^T\\x & tI\end{bmatrix}$. – Johan Löfberg Oct 2 '14 at 17:01