An upper bound on the expected value of a non-increasing convex function with convex marginal return functions

Abstract

In this note, we show that if a convex function is non-increasing and has a special property we call convex marginal return functions, an effective upper bound can be established using only two function evaluations. Further, we show that this bound can be refined in such a way that the number of function evaluations needed grows linearly with the number of refinements performed.