A Generalized Theorem of the Alternative for Certifiable Optimization with Redundant Constraints

Hanna Jiamei Zhang, Alan Papalia, Michael Everett, David M. Rosen

Keywords: Burer-Monteiro Factorization, Riemannian Staircase, Semidefinite Programming, Certifiable Optimization, Redundant Constraints

TL;DR: We prove a generalized theorem of the alternative for the Riemannian Staircase that does not require the LICQ, enabling certifiably correct low-rank optimization for SDPs with redundant constraints.

Abstract: Low-rank Semidefinite Programming via Burer–Monteiro (BM) factorization is a cornerstone of certifiably correct estimation in robotics. A key component is a theorem of the alternative that either certifies global optimality of a BM stationary point via positive semidefiniteness of a certificate matrix, or provides a second-order descent direction for continued search in a lifted dimension. However, current approaches require the linear independence constraint qualification (LICQ) to hold for the BM factorization, an assumption that necessarily fails when redundant constraints are added to tighten the semidefinite program (SDP) relaxation. Yet adding redundant constraints is a widely used technique for improving relaxation quality. We present a generalized theorem of the alternative that removes the LICQ requirement for BM factorized SDPs that does not depend on the LICQ. We show that global optimality can be certified via an eigenvalue maximization over the full multiplier set, and that this procedure preserves the certify-or-escape dichotomy: a nonnegative optimum certifies global optimality, while a negative optimum proves there are feasible points that reduce the objective.

Submission Number: 46