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

In this paper, we consider functions of the form phi(x, y) = f (x)g(y) over a box, where f (x), x is an element of R-n is a nonnegative monotone convex function with a power or an exponential form, and g(y), y is an element of 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.