Conic-closure equivalence of Eigen-CG and Boros–Hammer inequalities

Establish whether the convex conic closure generated by all Eigen-CG inequalities equals the convex conic closure generated by all Boros–Hammer inequalities for quadratically constrained quadratic programs reformulated over (x, X) with X − xx^T constrained via McCormick inequalities and eigenvector-derived cuts.

Background

Eigen-CG inequalities are obtained by applying Chvátal–Gomory rounding to eigenvector inequalities derived from the semidefinite constraint [1 xT; x X] ⪰ 0, yielding linear cuts valid for QPB_n. Boros–Hammer (BH) inequalities form a powerful, well-studied family valid for the Boolean quadric polytope (and, by Burer–Letchford’s result, also valid for QPB_n).

The paper defines nested subfamilies between BH and Eigen-CG and proves that the conic closures of BH, F1, and F2 are equal. It is conjectured that extending to the full Eigen-CG family does not enlarge the conic closure beyond BH, which would clarify that BH inequalities are as expressive as the entire Eigen-CG family in conic-closure terms.

References

We conjecture that the closure of $\mbox{Eigen-CG}$ is also equal to that of $ \mbox{BH}$, but we have not yet been able to formally prove this.

— Chvátal-Gomory Rounding of Eigenvector Inequalities for QCQPs  (2604.00932 - Dey et al., 1 Apr 2026) in Subsection: Contributions (Introduction)