On the Singularity Degree of Shor Relaxations for 0-1 Programs

Abstract

Shor relaxations are standard tools for nonconvex quadratic optimization, but they may fail Slater's condition. The singularity degree quantifies this degeneracy by the number of facial-reduction steps needed to reach the minimal face. We determine how the constraints defining a binary feasible set affect this quantity. For every nonempty binary set defined by linear equations and inequalities, the Shor relaxation has singularity degree at most one. It is zero precisely when the linear programming (LP) relaxation contains a point in the open cube that strictly satisfies every inequality, and is one otherwise. As an algorithmic consequence of this characterization, Slater's condition can be restored by solving an LP rather than an auxiliary semidefinite program (SDP). In sharp contrast, for every $n\geq1$, we construct a binary feasible set defined by quadratic equalities whose Shor relaxation has a positive semidefinite matrix variable of order $n+1$ and singularity degree $n$. Thus, singularity degree is uniformly bounded in the linearly constrained case but can grow linearly with the matrix order when quadratic defining constraints are used.