# Minimization problem involving the inverse of an affine matrix function

I want to minimize $$v^T (A+I+UQU^*)^{-1} v$$, subject to $$Q$$ and $$A$$ being positive semi-definite and $${\rm trace}(Q)<1$$. Here, $$v$$ is a given vector with unit norm, that is, $$\|v\|_2=1$$.

• Which variables here are kept fixed and which ones are you optimizing over? – user44191 Aug 13 '19 at 15:44
• the parameter to optimize is the matrix Q, its PSD symmetric and real with random value – hichem hb Aug 16 '19 at 9:42
• What does it mean to say that you are optimizing with respect to a parameter that has "random value"? – user44191 Aug 19 '19 at 3:58
• @user44191 all data are random variable so Q will be combination of random matrix am looking for this relation, i have try to solve the problem based on KKT theorem – hichem hb Aug 19 '19 at 13:22

Rephrasing slightly, given (symmetric) matrix $$\mathrm A \succeq \mathrm O_n$$, we have the following minimization problem in (symmetric) matrix $$\mathrm X \succeq \mathrm O_n$$

$$\begin{array}{ll} \text{minimize} & \mathrm v^\top \left( \mathrm A + \mathrm I_n + \mathrm U \mathrm X \mathrm U^\top \right)^{-1} \mathrm v \\ \text{subject to} & \mbox{tr} (\mathrm X) \leq 1\\ & \mathrm X \succeq \mathrm O_n\end{array}$$

Introducing a new optimization variable $$y \in \mathbb R$$ and rewriting in epigraph form,

$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & \mathrm v^\top \left( \mathrm A + \mathrm I_n + \mathrm U \mathrm X \mathrm U^\top \right)^{-1} \mathrm v \leq y \\ & \mbox{tr} (\mathrm X) \leq 1\\ & \mathrm X \succeq \mathrm O_n\end{array}$$

where the first inequality

$$y - \mathrm v^\top \left( \mathrm A + \mathrm I_n + \mathrm U \mathrm X \mathrm U^\top \right)^{-1} \mathrm v \geq 0$$

can be rewritten as the following linear matrix inequality (LMI) using the Schur complement

$$\begin{bmatrix} \mathrm A + \mathrm I_n + \mathrm U \mathrm X \mathrm U^\top & \mathrm v\\ \mathrm v^\top & y\end{bmatrix} \succeq \mathrm O_{n+1}$$

and, thus, we obtain the following semidefinite program (SDP) in $$\rm X$$ and $$y$$

$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & \begin{bmatrix} \mathrm A + \mathrm I_n + \mathrm U \mathrm X \mathrm U^\top & \mathrm v\\ \mathrm v^\top & y\end{bmatrix} \succeq \mathrm O_{n+1}\\ & \mbox{tr} (\mathrm X) \leq 1\\ & \mathrm X \succeq \mathrm O_n\end{array}$$

• Rodrigo de Azevedo can you please give me some reference to find my matrix X – hichem hb Sep 20 '19 at 10:30
• @hichemhb you can use CVX or CVXPY to solve the SDP numerically. – Rodrigo de Azevedo Sep 20 '19 at 10:34
• @ Rodrigo de Azevedo can you please give me your email i wanna to contact you ? – hichem hb Sep 21 '19 at 23:20
• it was about a problem in optimization that i find using KKT method – hichem hb Sep 21 '19 at 23:29
• @hichemhb Why not post a question on that problem on Math SE? The more (trained) eyes looking at it, the better. – Rodrigo de Azevedo Sep 21 '19 at 23:37