Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming

Abstract

An example of the quadratic assignment problem (QAP) is the facility location problem, in which n facilities are assigned, at minimum cost, to n sites. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the product of the flow and the distance between sites to which the facilities are assigned. Proving optimality of QAPs has been limited to instances having fewer than 20 facilities, largely because known lower bounds are weak. We compute lower bounds for a wide range of QAPs using a linear programming-based lower bound studied by Z. Drezner (1995). On the majority of QAPs tested, a new best known lower bound is computed. On 87% of the instances, we produced the best known lower bound. On several instances, including some having more the 20 facilities, the lower bound is tight. The linear programs, which can be large even for small QAPs, are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions. Attempts to solve these instances using the CPLEX primal simplex algorithm as well as the CPLEX barrier (primal-dual interior point) method were successful only for the smallest instances.