Sharp Singularity-Degree Bounds for Equality-Generated SDP-RLT Relaxations of Binary Programs

Abstract

Singularity degree is an important measure of semidefinite programming (SDP) degeneracy, but it is generally unavailable a priori from the problem data. We augment the Shor relaxation of nonempty binary sets $\{x\in\{0,1\}^n:Ax=b\}$ with the first-level Reformulation-Linearization Technique (RLT) equations generated by the defining linear equalities. For the resulting equality-generated SDP-RLT relaxation, we determine the exact worst-case singularity degree. If $\operatorname{rank}(A)=m$ and $0<m<n$, then the associated relaxation has singularity degree at most $\min\{m,n-m\}$, and this rank-nullity bound is attained for every possible rank in this range. Consequently, the worst-case singularity degree over this class is $\lfloor n/2\rfloor$ for $n\geq 2$. This is strikingly smaller than the sharp general bound $n$ for feasible SDP systems with matrix variables of order $n+1$ (Sturm, 2000, Example 2). Thus, for individual relaxations, rank and nullity provide an a priori bound on the otherwise inaccessible singularity degree and on the H\"older exponent in error bounds estimating distance to feasibility from constraint residuals.