When is the sum of a quasi-convex function and a convex function quasi-convex? 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?
This is not always true. In fact, constructing a counterexample is an exercise in Bauschke &amp; 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.
This is not always true. In fact, constructing a counterexample is an exercise in Bauschke &amp; 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.