Convex envelopes of products of convex and component-wise concave functions

Abstract

In this paper, we consider functions of the form $${\phi(x,y)=f(x)g(y)}$$ over a box, where $${f(x), x\in {\mathbb R}}$$ is a nonnegative monotone convex function with a power or an exponential form, and $${g(y), y\in {\mathbb R}^n}$$ is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.