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?