Chvátal-Gomory Rounding of Eigenvector Inequalities for QCQPs

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.

• The paper introduces the Eigen-CG family, integrating CG rounding with eigenvector inequalities to enhance relaxations for nonconvex QCQPs.
• It establishes that the convex conic closures of BH, F1, and F2 inequalities coincide, highlighting the expressive sufficiency of sparse cuts.
• Computational results show that incorporating sparse BH cuts into SDP and McCormick relaxations greatly reduces optimality gaps, while dense cuts yield minimal gains.

Chvátal-Gomory Rounding of Eigenvector Inequalities for Nonconvex QCQPs

Introduction and Theoretical Foundations

This work establishes a comprehensive analysis of cutting planes for nonconvex quadratically constrained quadratic programming (QCQP) with a focus on a novel family of valid inequalities termed Eigen-CG inequalities. By integrating Chvátal-Gomory (CG) rounding with classical eigenvector-based inequalities, the study extends the polyhedral theory of relaxations for QCQPs. The central methodological innovation is the derivation of these inequalities by CG-rounding of eigenvector inequalities followed by lifting from Boolean to the continuous $[0,1]^n$ domain, leveraging results from Burer and Letchford.

Three nested families of inequalities are defined:

• The Boros-Hammer (BH) inequalities (integral parametrization).
• Two relaxations ($\mathcal{F}_1$, $\mathcal{F}_2$) allowing progressively less restrictive coefficient rounding.
• The full Eigen-CG family, supporting arbitrary real-valued parametrization.

The authors establish strict containment between these families: $\mathrm{BH} \subsetneq \mathcal{F}_1 \subsetneq \mathcal{F}_2 \subsetneq \mathrm{Eigen\mhyphen CG}$. Notably, the convex conic closures of BH, $\mathcal{F}_1$, and $\mathcal{F}_2$ coincide, demonstrating that the extended rounding freedom does not enlarge the closure beyond what is provided by BH inequalities. Computational and combinatorial arguments support the conjecture that Eigen-CG inequalities as a whole have the same closure as BH, although a complete proof is left open.

Structural Results and Limitations

The study advances several key theoretical contributions:

• Equivalence of Closures: The convex conic closure results reinforce that BH inequalities are in a precise sense maximally expressive among all cuts derivable by CG-rounding of quadratic eigenvector inequalities under the Burer-Letchford lifting.
• Facet Enumeration and Expressiveness: For $n \leq 5$ (i.e., Boolean Quadric Polytope of dimension at most 5), all facets are BH (and thus Eigen-CG) inequalities. For $n=6$, exhaustive enumeration confirms that the vast majority of facets (over 96%) are outside the Eigen-CG (or BH) representable class, confirming both the power and the boundary of these families. Explicit proof methods are devised for certifying non-Eigen-CG facets, exploiting sign patterns and coefficient identities.
• Density and Depth of Cuts: Theoretical bounds show that the depth (or efficacy) of any Eigen-CG inequality degrades rapidly with the cardinality (support) of the underlying eigenvector, especially when imposed atop semidefinite plus McCormick relaxations. Dense inequalities unambiguously lose separation power; this is analytically quantified by upper bounds that scale as $O(1/s)$ for sparsity $s$.

Computational Methodology

The empirical analysis uses BIQ instances to assess the tightening power of Eigen-CG/BH facet inequalities beyond standard SDP+McCormick or triangle relaxations. Separation of sparse inequalities (facets of $\mathcal{F}_1$0, $\mathcal{F}_1$1) is achieved via enumeration and lookup, while denser inequalities from larger submatrices require a dedicated BH separator, solved as an integer quadratic program with sparsity-inducing constraints and time limits.

Experiments evaluate:

• The effect of exclusively using classical relaxations (McCormick, triangle, SDP).
• Incremental strengthening by incorporating violated $\mathcal{F}_1$2 and $\mathcal{F}_1$3 facets iteratively.
• The impact of high-density BH inequalities generated via nonconvex optimization.

Numerical Results

Results show:

• Adding sparse BH (facet) inequalities on top of SDP+McCormick and triangle relaxations very efficiently reduces or closes optimality gaps in standard BIQ test sets.
• The marginal improvement from moving beyond triangle inequalities to higher-order ($\mathcal{F}_1$4, $\mathcal{F}_1$5) and then even sparser BH inequalities is significant, but fast-diminishing.
• Inclusion of denser BH/Eigen-CG cuts yields minimal gains in the presence of an SDP constraint, incurring heavy computational costs for only incremental strengthening.
• In relaxations without the SDP constraint, denser inequalities contribute more noticeably, but the overall relaxation remains looser compared to when the SDP constraint is present.

Implications and Future Directions

The findings have several important implications:

• Practical impact: The results confirm that modern SDP-based relaxations for QCQPs can be significantly strengthened by systematically adding sparse facet (BH/Eigen-CG) inequalities beyond the triangle class. This has direct computational benefits for solving nonconvex binary quadratic optimization problems and related combinatorial instances.
• Theoretical insight: The closure results clarify the maximal expressive power of BH-like rounds, linking polynomial-time separability (for structured subfamilies) with their sufficiency at lower dimensionality and clear insufficiency as $\mathcal{F}_1$6 grows.
• Algorithmic design: The evidence points toward a sparsity principle—focusing algorithmic development on the efficient discovery and separation of sparse, structured cuts (small support inequalities), since the inclusion of dense, high-order Eigen-CG inequalities is both computationally and polyhedrally inefficient.
• The conjectured equivalence of Eigen-CG and BH closures remains open, and resolving this would solidify understanding of general quadratic lift-and-project hierarchies in the continuous domain.

Conclusion

This work provides a rigorous structural, algorithmic, and computational analysis of CG-rounded eigenvector inequalities for nonconvex QCQPs, highlighting the expressive sufficiency and concrete limitations of the BH/Eigen-CG family. The theoretical closure equivalence, explicit non-facet coverage for higher dimensional polytopes, and the pronounced empirical enhancement arising from sparse facet separation collectively emphasize that future advances will benefit from focusing on the selective addition of sparse, well-structured valid inequalities in extended SDP relaxations rather than reliance on high-density eigenvector rounds. These insights have direct implications for the design of next-generation SDP and polyhedral cut-generation algorithms for broad classes of continuous and discrete quadratic optimization.