Error bounds for rank-one DNN reformulation of QAP and DC exact penalty approach (2403.11210v1)

Abstract: This paper concerns the quadratic assignment problem (QAP), a class of challenging combinatorial optimization problems. We provide an equivalent rank-one doubly nonnegative (DNN) reformulation with fewer equality constraints, and derive the local error bounds for its feasible set. By leveraging these error bounds, we prove that the penalty problem induced by the difference of convexity (DC) reformulation of the rank-one constraint is a global exact penalty, and so is the penalty problem for its Burer-Monteiro (BM) factorization. As a byproduct, we verify that the penalty problem for the rank-one DNN reformulation proposed in \cite{Jiang21} is a global exact penalty without the calmness assumption. Then, we develop a continuous relaxation approach by seeking approximate stationary points of a finite number of penalty problems for the BM factorization with an augmented Lagrangian method, whose asymptotic convergence certificate is also provided under a mild condition. Numerical comparison with Gurobi for \textbf{131} benchmark instances validates the efficiency of the proposed DC exact penalty approach.