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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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    $\begingroup$ Which variables here are kept fixed and which ones are you optimizing over? $\endgroup$
    – user44191
    Aug 13, 2019 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, 2019 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, 2019 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, 2019 at 13:22

1 Answer 1

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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 $\endgroup$
    – hichem hb
    Sep 20, 2019 at 10:30
  • $\begingroup$ @hichemhb you can use CVX or CVXPY to solve the SDP numerically. $\endgroup$ Sep 20, 2019 at 10:34
  • $\begingroup$ @ Rodrigo de Azevedo can you please give me your email i wanna to contact you ? $\endgroup$
    – hichem hb
    Sep 21, 2019 at 23:20
  • $\begingroup$ it was about a problem in optimization that i find using KKT method $\endgroup$
    – hichem hb
    Sep 21, 2019 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. $\endgroup$ Sep 21, 2019 at 23:37

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