Consider the following non-convex problem:
\begin{equation*} \begin{aligned} & \text{maximize} & & f(X) \\ & \text{subject to} & & f(X)\le b\\ &&& A_kX = c_k, \ k=1,..,n\\ &&& X \succeq 0. \end{aligned} \end{equation*}
Where $f(X)$ is a convex function and $A_kX=c_k$ are linear constraints. In addition, we know that there is unique solution $X^*$ such that $f(X^*)=b$. In other words, if we indeed succeed in maximizing $f(X)$ under these constraints then we have found the unique solution $X^*$.
My questions are:
1) Can this problem be in principle be solved? by solved I mean that we are guranteeed to find the solution $X^*$, that is, maximizing $f(X)$ under the given constraints.
2) What algorithm is reccomended for this problem?
Thanks.
In principle yes, but it is non-convex so in practice hard. Trivial reformulation is to simply define $X$ as $RR^T$, and now you have a standard non-convex program which you can throw your favorite global non-convex solver on. Alternatively, upper bound the objective with a simple concave function, solve the relaxation, and then proceed with a spatial branch-and-bound strategy. Not saying these are prefered or good approaches, just answering the question whether it is possible in principle.