A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP

Abstract

Splitting methods in optimization arise when one can divide an optimization problem into two or more simpler subproblems. They have proven particularly successful for relaxations of problems involving discrete variables. We revisit and strengthen splitting methods for solving doubly nonnegative relaxations of the particularly difficult, NP-hard quadratic assignment problem. We use a modified restricted contractive splitting method approach. In particular, we show how to exploit redundant constraints in the subproblems. Our strengthened bounds exploit these new subproblems and new dual multiplier estimates to improve on the bounds and convergence results in the literature. Summary of Contribution: In our paper, we consider the quadratic assignment problem (QAP). It is one of the fundamental combinatorial optimization problems in the fields of optimization and operations research and includes many fundamental applications. We revisit and strengthen splitting methods for solving doubly nonnegative (DNN) relaxation of the QAP. We use a modified restricted contractive splitting method. We obtain strengthened bounds from improved lower and upper bounding techniques, and in fact, we solve many of these NP-hard problems to (provable) optimality, thus illustrating both the strength of the DNN relaxation and our new bounding techniques.