Maximal quadratic-free sets

TL;DR

This paper introduces a novel method for constructing maximal quadratic-free sets, enhancing the intersection cut framework for quadratic inequalities and improving vertex separation in LP-based quadratic problem solving.

Contribution

It is the first work to provide a systematic approach for maximal quadratic-free sets, advancing cutting plane techniques for quadratic inequalities.

Findings

Constructed maximal quadratic-free sets for general quadratic inequalities.

Guaranteed efficient separation of vertices in LP-based quadratic problems.

First known method for maximal quadratic-free sets.

Abstract

The intersection cut paradigm is a powerful framework that facilitates the generation of valid linear inequalities, or cutting planes, for a potentially complex set S. The key ingredients in this construction are a simplicial conic relaxation of S and an S-free set: a convex zone whose interior does not intersect S. Ideally, such S-free set would be maximal inclusion-wise, as it would generate a deeper cutting plane. However, maximality can be a challenging goal in general. In this work, we show how to construct maximal S-free sets when S is defined as a general quadratic inequality. Our maximal S-free sets are such that efficient separation of a vertex in LP-based approaches to quadratically constrained problems is guaranteed. To the best of our knowledge, this work is the first to provide maximal quadratic-free sets.