Suppose I have a problem of the form
subject to some (convex) inequality constraints and some affine equality constraints, and where and are known to be convex, and is some known constant. We can also assume that the feasible set is compact (in addition to being convex).
The main challenge here is when is not convex (otherwise it is just a convex problem), and also we don't have an easy escape like and are log convex, for example.
In my specific situation, I also know that are of the form
for some (known) nondecreasing and integrable (but not necessarily continuous) functions and for some real numbers .
Are there any easy ways to tackle this sort of problem? I am hoping to find ways that would be computationally not much more difficult than convex optimization, but I am not sure if this is possible... it seems like even the simpler problem of minimizing is (surprisingly) difficult...