Aggregations of quadratic inequalities and hidden hyperplane convexity

TL;DR

This paper investigates when aggregations of quadratic inequalities can describe the convex hull of a set, introducing the concept of hidden hyperplane convexity (HHC) to generalize and identify new classes of such sets.

Contribution

It introduces the notion of hidden hyperplane convexity (HHC) and shows it ensures aggregations define convex hulls, generalizing previous results and identifying new classes of sets.

Findings

HHC relates to hidden convexity of quadratic maps.

Positive definite linear combinations with HHC define convex hulls.

New results for closed quadratic inequalities without topological assumptions.

Abstract

We study properties of the convex hull of a set $S$ described by quadratic inequalities. A simple way of generating inequalities valid on $S$ is to take a nonnegative linear combinations of the defining inequalities of $S$. We call such inequalities aggregations. Special aggregations naturally contain the convex hull of $S$, and we give sufficient conditions for such aggregations to define the convex hull. We introduce the notion of hidden hyperplane convexity (HHC), which is related to the classical notion of hidden convexity of quadratic maps. We show that if the quadratic map associated with $S$ satisfies HHC, then the convex hull of $S$ is defined by special aggregations. To the best of our knowledge, this result generalizes all known results regarding aggregations defining convex hulls. Using this sufficient condition, we are able to recognize previously unknown classes of sets where aggregations lead to convex hull. We show that the condition known as positive definite linear combination together with hidden hyerplane convexity is a sufficient condition for finitely many aggregations to define the convex hull. All the above results are for sets defined using open quadratic inequalities. For closed quadratic inequalities, we prove a new result regarding aggregations giving the convex hull, without topological assumptions on $S$.