Almost all the semidefinite programming paper I found has a linear objective function, which is $\mbox{tr}(CX)$. Why?
For example, in this paper and Wikipedia, the objective function is always $\mbox{tr}(CX)$, where $C$ is a known matrix. The goal is to minimize $\mbox{tr}(CX)$, such that $X\succeq0$ and satisfy some linear constraints. I wonder what is the difficulty to generalize the objective function $\mbox{tr}(CX)$ to general cases, e.g. convex functions? Such as strictly convex smooth objective function $\ell(X)=\mbox{tr}(CXX)$, where $C\succ0$? Or $\ell(X)=\mbox{tr}(CXX) -\sum_{i=1}^{n}\log(X_{ii})$?