As I am rather unfamiliar with optimization and convex programming my question might be a bit naive.
Let's start with a very simple semidefinite program $$\min \, \, tr(X)$$ $$ s.t.\, X \succeq 0 \quad \text{and} \quad \langle X, A_k\rangle = b_k, \quad k\in I$$ which is concerned with minimizing the trace over the intersection of an affine set with the cone of positive matrices.
If we replace $ tr(X)=\tau_1(X)$ by $\tau_\infty(X)$ (or any $\tau_p$ in between) where $$ \tau_p(X) := \|(X_{i,i})_{i=1}^d\|_{\ell^p} $$ the $p$-norm over the diagonal we loose linearity and the resulting problem does not qualify as a semidefinite program any more.
My question now is: What is the best way to solve the modified problem? Would one just use a general purpose solver or are there specialized methods for this type of problem?