Consider the following optimization problem:
\begin{equation*} \begin{split} \max_{X,Y} \;& \mathrm{Tr}[A Y] + \mathrm{Tr}[(B X B^\top)^{1/2}]\\ \mbox{s.t.} \; & Y = C - F^\top B^\top (X + D)^{-1} B F\\ & X \succeq 0, Y\succeq 0, \end{split} \end{equation*} where $A, B, C, D$ and $F$ are known matrices of appropriate dimension. Moreover, $A, B$, and $D$ are positive semidefinite.
This problem is intractable due to the inverse in the first constraint and the square root in the second trace in the objective. My question is, is it possible to reformulate this problem into an SDP?
Here are the reformulations I tried:
- I replaced the first equality with an inequality: \begin{equation*} Y \preceq C - F^\top B^\top (X + D)^{-1} B F. \end{equation*}
(Not sure if this is allowed) Then, I used the Schur complement lemma to write the inequality constraint as the following LMI constraint: \begin{equation*} \begin{bmatrix} C - Y & F^\top B^\top \\ B F & X + D \end{bmatrix} \succeq 0, \end{equation*} with \begin{equation*} X + D\succeq 0. \end{equation*}
- For the objective function, I've introduced a new variable $U \succeq 0$ and a new constraint \begin{equation*} (BXB^\top)^{1/2} \succeq U, \end{equation*} which again can be written as the following LMI constraint: \begin{equation*} \begin{bmatrix} B X B^\top & U \\ U & I \end{bmatrix}\succeq 0. \end{equation*}
As a result, the final form of the problem I obtained is as follows: \begin{equation*} \begin{split} \max_{X,Y, U} \;& \mathrm{Tr}[A Y] + \mathrm{Tr}[U]\\ \mbox{s.t.} \; & \begin{bmatrix} Y - C & FB \\ B^\top F^\top & X + D \end{bmatrix} \succeq 0\\ & \begin{bmatrix} B X B^\top & U \\ U & I \end{bmatrix}\succeq 0. \\ &X + D\succeq 0\\ & X \succeq 0, Y\succeq 0, U\succeq 0. \end{split} \end{equation*}
I would like to know whether my reformulations presented above are valid.