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

摘要

In this paper, we consider functions of the form φ(x, y) = f(x)g(y) overabox, where f(x),x g R is a nonnegative monotone convex function with a power or an exponential form, and g(y), y ∈ 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.