How to deal with a convex constraint

I want to deal with a convex constraint \begin{align} F(P)=P^{H}AP_{0}+P_{0}^{H}AP-P_{0}^{H}AP_{0}\succeq 0 \end{align} where $(\cdot)^{H}$ represents Hermitian transpose, $A$ is a positive definite matrix, $P_{0}$ is a given square matrix and $P$ is also a square matrix. Since this constraint can be represented alternatively as \begin{align} v^{H}(P^{H}AP_{0}+P_{0}^{H}AP-P_{0}^{H}AP_{0})v \geq 0,\quad \forall v\in C^{n} \end{align} which is a affine function of $P$, we know that this is a convex constraint. However, I'm not sure how to deal with this constraint in a convex optimization problem. Any suggestions will be highly appreciated, thank you.

• Define "deal with". What are you trying to do? What is the full model you're trying to solve? This is just a linear matrix inequality, and there is a wealth of information on the internet about how to solve problems involving LMIs. – Michael Grant Dec 1 '15 at 14:39