Benign Degeneracy for First-Order Methods in Exact Boolean Moment-SOS Relaxations

Abstract

Pseudo-Boolean optimization -- minimizing a polynomial over $\{\pm 1\}^n$ -- includes MAX-CUT, binary quadratic optimization, and Ising models. The moment-sum-of-squares (Moment-SOS) hierarchy provides certified semidefinite relaxations that are often empirically exact at low order. We ask: once a relaxation is exact, how difficult is the resulting semidefinite program (SDP) to solve with first-order methods? Exactness does not imply regularity. At every order $d \ge 2$, a unique rank-one moment optimum is primal degenerate; moreover, if one optimal SOS certificate is strictly complementary, then the certificate set is positive-dimensional and contains singular boundary points. Thus first-order methods may converge to singular endpoints, around which pointwise analyses that assume regularity at the limit cannot establish a local linear convergence rate. We show that, for exact Boolean Moment-SOS relaxations in their canonical quotient form, this degeneracy is nevertheless benign. We develop an abstract mechanism, which we call residual-field rigidity, that propagates regularity from the relative interior of the optimal set to its singular boundary. For Boolean Moment-SOS, unique rank-one exactness and the existence of one strictly complementary certificate suffice to trigger this mechanism, yielding a uniform error bound around the entire optimal set. Consequently, exact ADMM, exact sGS-ADMM, and fixed-step PDHG converge eventually linearly from every initialization, even when their limiting certificate is singular. Finally, outside a lower-dimensional semialgebraic exceptional set of Boolean objectives, these hypotheses hold simultaneously at every uniquely rank-one exact order.