Dominance of Eigen-CG inequalities by Boros–Hammer combinations

Prove that for every real vector (v0, v) in R^{n+1}, the Eigen-CG inequality E-CG(v0, v)—obtained by Chvátal–Gomory rounding of the eigenvector inequality associated with [1 x^T; x X] ⪰ 0—is implied by a nonnegative linear combination of Boros–Hammer inequalities.

Background

The authors show that, for the restricted family F2, every Eigen-CG inequality is implied by a nonnegative combination of BH inequalities, hence F2’s conic closure equals that of BH. They conjecture extending this implication to all Eigen-CG inequalities.

If true, this would imply that the conic closure of the full Eigen-CG family coincides with BH’s conic closure, reinforcing the expressive completeness of BH inequalities for this setting.

References

For any $(v_0, v)\in \mathbb{R}{n+1}$, the inequality defined by $\mbox{E-CG}(v_0,v)$ is implied by a nonnegative combination of BH inequalities. To date, every computational test we have devised suggests that this conjecture is true, but we have not been able to find a proof.

— Chvátal-Gomory Rounding of Eigenvector Inequalities for QCQPs  (2604.00932 - Dey et al., 1 Apr 2026) in Conjecture (label “conjecture”), Subsection: Closure comparison of restricted families