# Weighted PSD matrix minimization

Suppose that $D\in \mathbb{R}^{d \times d}$ is a positive diagonal matrix, $A\in \mathbb{R}^{d \times d}$ is a real matrix (not necessary symmetric), $\Omega$ is the symmetric positive semi-definite matrix set. We want to solve the following problem: \begin{align} \min_{\Sigma \in \Omega}\|D\Sigma - A\|_F, \end{align} where $\|\cdot \|_F$ is the Frobenius norm of matrix. Is there any closed-form or numerical solution to this problem?

• It can be easily solved with no assumption on $D$ (Projected Gradient Descent). I'm trying to figure out how to take advantage of $D$ (Besides the obvious in the Gradient Descent step). – Royi Sep 10 '18 at 22:28