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, $\|v\|_2=1$.
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2$\begingroup$ Which variables here are kept fixed and which ones are you optimizing over? $\endgroup$– user44191Aug 13, 2019 at 15:44
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$\begingroup$ the parameter to optimize is the matrix Q, its PSD symmetric and real with random value $\endgroup$– hichem hbAug 16, 2019 at 9:42
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$\begingroup$ What does it mean to say that you are optimizing with respect to a parameter that has "random value"? $\endgroup$– user44191Aug 19, 2019 at 3:58
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$\begingroup$ @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 $\endgroup$– hichem hbAug 19, 2019 at 13:22
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1 Answer
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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 $y$
$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & \begin{bmatrix} \mathrm A + \mathrm I_n + \mathrm U \mathrm X \mathrm U^\top & \mathrm v\\ \mathrm v^\top & y\end{bmatrix} \succeq \mathrm O_{n+1}\\ & \mbox{tr} (\mathrm X) \leq 1\\ & \mathrm X \succeq \mathrm O_n\end{array}$$
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$\begingroup$ Rodrigo de Azevedo can you please give me some reference to find my matrix X $\endgroup$ Sep 20, 2019 at 10:30
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$\begingroup$ @hichemhb you can use CVX or CVXPY to solve the SDP numerically. $\endgroup$ Sep 20, 2019 at 10:34
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$\begingroup$ @ Rodrigo de Azevedo can you please give me your email i wanna to contact you ? $\endgroup$ Sep 21, 2019 at 23:20
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$\begingroup$ it was about a problem in optimization that i find using KKT method $\endgroup$ Sep 21, 2019 at 23:29
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$\begingroup$ @hichemhb Why not post a question on that problem on Math SE? The more (trained) eyes looking at it, the better. $\endgroup$ Sep 21, 2019 at 23:37