Chv\'atal-Gomory Rounding of Eigenvector Inequalities for QCQPs

TL;DR

This paper introduces Eigen-CG inequalities for nonconvex QCQPs, analyzing their properties, limitations, and effectiveness, especially emphasizing the importance of sparsity for practical improvements.

Contribution

It develops a new class of inequalities derived via Chvátal-Gomory rounding, analyzes their structure, and proposes a computational strategy for sparse cuts to improve dual bounds.

Findings

Dense Eigen-CG inequalities are ineffective with standard relaxations.

Sparse Eigen-CG inequalities can significantly improve dual bounds.

The convex conic closure of certain subfamilies coincides with Boros-Hammer inequalities.

Abstract

We introduce and analyze a class of valid inequalities for nonconvex quadratically constrained optimization problems (QCQPs) which we call Eigen-CG inequalities. These inequalities are obtained by applying a Chv\'atal-Gomory (CG) rounding to the well-known eigenvector inequalities for QCQPs, and transferring binary-valid inequalities to the continuous setting via a result of Burer and Letchford (2009). We define three nested subfamilies and prove that they are strictly contained in one another. However, we show that the convex conic closure of two of these subfamilies is equal and, in fact, coincides with the Boros-Hammer inequalities -- a powerful family of inequalities that include, in particular, the triangle and McCormick inequalities. Using this CG perspective, we also prove that dense Eigen-CG inequalities are ineffective when used with the standard SDP+McCormick relaxation. This provides a complementary perspective on what is observed in practice: that sparse inequalities are impactful. Finally, based on these insights, we develop a computational strategy to find sparse Eigen-CG cuts and verify their effectiveness in nonconvex QCQP instances. Our results confirm that density quickly degrades effectiveness, but that including sparse inequalities beyond triangle inequalities can provide significant improvements in dual bounds.