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, $\|v\|_2=1$.

  • 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.