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

2$\begingroup$ Which variables here are kept fixed and which ones are you optimizing over? $\endgroup$ – user44191 Aug 13 '19 at 15:44

$\begingroup$ the parameter to optimize is the matrix Q, its PSD symmetric and real with random value $\endgroup$ – hichem hb Aug 16 '19 at 9:42

$\begingroup$ What does it mean to say that you are optimizing with respect to a parameter that has "random value"? $\endgroup$ – user44191 Aug 19 '19 at 3:58

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

$\begingroup$ Rodrigo de Azevedo can you please give me some reference to find my matrix X $\endgroup$ – hichem hb Sep 20 '19 at 10:30

$\begingroup$ @hichemhb you can use CVX or CVXPY to solve the SDP numerically. $\endgroup$ – Rodrigo de Azevedo Sep 20 '19 at 10:34

$\begingroup$ @ Rodrigo de Azevedo can you please give me your email i wanna to contact you ? $\endgroup$ – hichem hb Sep 21 '19 at 23:20

$\begingroup$ it was about a problem in optimization that i find using KKT method $\endgroup$ – hichem hb Sep 21 '19 at 23:29

$\begingroup$ @hichemhb Why not post a question on that problem on Math SE? The more (trained) eyes looking at it, the better. $\endgroup$ – Rodrigo de Azevedo Sep 21 '19 at 23:37