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.

Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

“…Hence designing a model that has strict feasibility is important. However in cases where the absence of strict feasibility is inevitable (e.g., SDP relaxations of discrete optimization problems), facial reduction can be performed to regularize the model, e.g., see [19]. Typically, strict feasibility for LPs is less emphasized and many algorithms show strong numerical performances without this assumption.…”

On solving the MAX-SAT using sum of squares

“…However, for medium and large size instances, interior point methods suffer from a large computation time and memory demand, which has recently motivated researchers to consider first order methods, such as the PRSM. For recent applications of PRSM to SDP, see e.g., [14,19]. Section 7.2 and Section 7.3 provide details on obtaining valid upper and lower bounds, from the output of the PRSM algorithm.…”

“…Consider the clauses C j of length three, which can all be factorized as g j (δ)f j (Y ) = 0, as shown in (19). For all j such that g j (δ) = 0, let B j be the set of the two subsets appearing in the intersection of f j (Y ) and F .…”