maximizing a function of a positive semi-definite matrix with bounded trace

Question

I need to maximize a function $f(A)$ where $A$. With the constraints that $A$ is positive definite and has a trace $tr(A) \leq K$. $tr(A)=K$ will work for my problem too. I can differentiate towards $A_{ij}$ and use Lagrange multipliers for the trace constraint, but how do I put a constraint on the positive definitenes?

Answer 1

If you search the maximum of $f$ and not its upper bound, then the constraints associated to "$A$ positive definite" are open except the condition $A\in S$, where $S$ is the vector space of symmetric matrices. Then, in a first time, you must solve $2$ problems concerning $f:S\rightarrow \mathbb{R}$.

1. $\max_S(f)$. The equation to solve is: for every $H\in S$, $Df_A(H)=0$.

2. $\max_S(f)$ under the condition $trace(A)-K=0$. The Lagrange's equation to solve is: there is a real $\lambda$ s.t.for every $H\in S$, $Df_A(H)+\lambda trace(H)=0$.