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 pdimensional 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!

$\begingroup$ I think you meant beta=(beta+)(beta) and beta=(beta+)+(beta). $\endgroup$– Cristóbal GuzmánApr 5, 2014 at 11:13

$\begingroup$ Have you tried to formulate the optimality system, e.g. by using this: en.wikipedia.org/wiki/… $\endgroup$– DirkFeb 2, 2015 at 8:36
1 Answer
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 rewrite 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

1$\begingroup$ Why are you introducing $B$ and $F$? Introduce a scalar $t$ and simply minimize $t$ subject to $\begin{bmatrix}D+T & x^T\\x & tI\end{bmatrix}$. $\endgroup$ Oct 2, 2014 at 17:01

$\begingroup$ EDIT: Just saw this was an old question. Didn't mean to bring it up from the dead $\endgroup$ Oct 2, 2014 at 17:09