On obtaining the convex hull of quadratic inequalities via aggregations

TL;DR

This paper investigates how to derive the convex hull of sets defined by multiple quadratic inequalities, extending previous results from two inequalities to three or more under certain conditions.

Contribution

It demonstrates that the convex hull of three quadratic inequalities can be obtained through aggregations under technical assumptions, unlike the case with four or more inequalities.

Findings

Convex hull of three quadratics can be derived via aggregations under specific conditions.

Counterexamples show the approach likely fails with four or more inequalities.

Extension of aggregation methods from linear to quadratic inequalities.

Abstract

A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran showed that the convex hull of the set is given by at most two aggregated inequalities. In this work, we study the case of a set described by three or more quadratic inequalities. We show that, under technical assumptions, the convex hull of a set described by three quadratic inequalities can be obtained via (potentially infinitely many) aggregated inequalities. We also show, through counterexamples, that it is unlikely to have a similar result if either the technical conditions are relaxed, or if we consider four or more inequalities.