TITLE: Minimization problem involving the inverse of an affine matrix function
QUESTION [1 upvotes]: 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, $\|v\|_2=1$.

REPLY [6 votes]: 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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