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arXiv:2604.00932

Chvátal-Gomory Rounding of Eigenvector Inequalities for QCQPs

Santanu S. Dey, Nan Jiang, Aleksandr M. Kazachkov, Andrea Lodi, Gonzalo Muñoz

Published Apr 1, 2026• 0 citations
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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 Chvátal-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.

DOI
10.48550/arxiv.2604.00932

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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 Chvátal-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.
The integer quadratic moment hull is the closed convex hull of the pairs (z,zz⊤)(z,zz^{\top})(z,zz⊤) with z∈Znz\in\mathbb{Z}^nz∈Zn. We prove that one round of Chvátal-Gomory (CG) cuts applied to its basic semidefinite relaxation recovers this hull exactly if and only if n≤6n\le6n≤6. Our approach identifies its facets with perfect Delaunay polyhedra and characterizes their recovery through depth, the gap between lattice and continuous minima of a defining quadratic after normalization. A facet inequality is valid for the rank-one CG closure exactly when its depth is less than one. We also analyze the complexity of deciding membership in the CG closure and finding violated inequalities. Every rational point outside the CG closure admits a violated CG inequality whose encoding length is polynomial in the input length. Membership is strongly coNP-complete and exact separation is strongly NP-hard. The same reduction proves strong NP-hardness of quadratic split-inequality separation, resolving a conjecture of Burer and Letchford.
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ABSTRACT Quadratically constrained quadratic programming (QCQP) problems appear in a wide range of engineering fields, including computer science, communication engineering, and finance. A key difficulty in solving these problems lies in efficiently finding global solutions, especially for large-scale and nonconvex instances. To address this, Wen & Yin (2013) proposed a method that reformulates semidefinite programming (SDP) relaxations of QCQPs into a nonlinear and nonconvex low-rank problem. This reformulated problem can be efficiently solved using a curvilinear search method combined with Barzilai-Borwein (BB) steps, known as the CSBB algorithm. In this study, we compare two approaches for solving QCQPs: the conventional convex SDP relaxation and the nonconvex low-rank reformulation introduced by Wen and Yin. We propose a set of general QCQP models that are compatible with the low-rank framework and conduct a series of numerical experiments to evaluate their performance. This study evaluates performance using several classes of NP-hard problems, including Binary Integer Quadratic (BIQ) problems, Max-Cut problems, Boolean Least Squares (BLS) problems, and 0-1 Quadratic Knapsack Problems (QKP). The results demonstrate that the low-rank approach offers competitive performance and shows strong potential for solving large-scale QCQPs more efficiently than traditional convex methods.
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