On obtaining the convex hull of quadratic inequalities via aggregations\ast

Abstract

A classical approach for obtaining valid inequalities for a set involves the analysis of relaxations constructed using aggregations of the inequalities that describe such a 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 [IMA J. Math. Control Inform., 26 (2009), pp. 417--450] 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 such as a result does not hold either if the technical conditions are relaxed or if we consider four or more inequalities.