Convex envelopes for ray-concave functions

TL;DR

This paper introduces a new class of functions called ray-concave functions and provides methods to compute their convex envelopes over polytopes, offering new insights and explicit formulas for functions relevant in probability and optimization.

Contribution

The paper defines ray-concave functions, establishes conditions for their convex envelopes, and derives explicit formulas, including a novel envelope for a probability-related function.

Findings

Closed-form convex envelopes for ray-concave functions over polytopes

New perspective on known convex envelopes

Explicit convex envelope for a probability-related function

Abstract

Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable number of functions that appear in practice, and thus obtain tight and tractable approximations to challenging problems. We contribute to this line of work by considering a family of functions that, to the best of our knowledge, has not been considered before in the literature. We call this family ray-concave functions. We show sufficient conditions that allow us to easily compute closed-form expressions for the convex envelope of ray-concave functions over arbitrary polytopes. With these tools, we are able to provide new perspectives to previously known convex envelopes and derive a previously unknown convex envelope for a function that arises in probability contexts.