Optimizing convex functions over nonconvex sets

TL;DR

This paper develops strong linear inequalities to describe the convex hull of sets defined by quadratic functions over nonconvex domains, enabling more efficient optimization over such sets.

Contribution

It introduces new linear inequalities that characterize the convex hull of quadratic sets outside convex regions, with efficient separation methods.

Findings

Derived strong linear inequalities for quadratic sets

Characterized convex hulls with separable inequalities

Applicable to convex and certain nonconvex sets

Abstract

In this paper we derive strong linear inequalities for sets of the form {(x, q) \in Rd \times R : q \geq Q(x), x \in Rd - int(P)}, where Q(x) : Rd \rightarrow R is a quadratic function, P \subset Rd and "int" denotes interior. Of particular but not exclusive interest is the case where P denotes a closed convex set. In this paper, we present several cases where it is possible to characterize the convex hull by efficiently separable linear inequalities.