When is the sum of a quasi-convex function and a convex function quasi-convex?

Question

I know that the sum of a convex function and a quasi-convex function is not necessarily convex, it is easy to construct counter-examples like $f(x)=-x$ and $g(x)=x-1/2|x|$.

However, I have a setting (dynamic programming with minimization function involved, so I cannot simplify it easily) where I strongly assume that my sum of convex function and quasi-convex function is a quasi-convex function.

Within my search for this, I am a bit stuck on where to start. Are there some well-known properties on one or both of my functions that would make the sum a quasi-convex function?

Answer 1

This is not always true. In fact, constructing a counterexample is an exercise in Bauschke & Combettes' book (volume 2), Exercise 10.15. Without seeing your particular functions, it is hard to say whether or not their sum will be quasiconvex.