# Maximize Trace as standard semidefinite optimization

Let $A$ be a symmetric matrix and $X$ a symmetric positive definite matrix, then the following standard semidefinite optimization problem is convex:

min $tr (AX)$ subject to $X>0$

Now I wonder if $tr ((-A)X)$ is convex or concave?

The reason is the following. I want to solve: max $tr (AX)$ subject to $X>0$

Can that be written as: min $tr ((-A)X)$ subject to $X>0$ ? Is that convex and hence solvable?

• Any linear function of $X$ is convex. – littleO Oct 11 '13 at 11:02

It is still convex. Note that for semi-definite optimization, the requirement is that the objective function and constraints (except the semi-definite constraint) are linear in terms of the entries of $\mathbf{X}$.