A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

We apply the level-3 Reformulation Linearization Technique (RLT3) to the Quadratic Assignment Problem (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm, based on this RLT3 formulation. This algorithm is not guaranteed to calculate the RLT3 lower bound exactly, but approximates it very closely and reaches it in some instances. For Nugent problem instances up to size 24, our RLT3-based lower bound calculation solves these problem instances exactly or serves to verify the optimal value. Calculating lower bounds for problems sizes larger than size 25 still presents a challenge due to the large memory needed to implement the RLT3 formulation. Our presentation emphasizes the steps taken to significantly conserve memory by using the numerous problem symmetries in the RLT3 formulation of the QAP. The authors are grateful to management and consulting staff of the San Diego Supercomputing Center for the invaluable computational resources and guidance they provided for this work. We thank Professor Matthew Saltzman of the Mathematics Department at Clemson University for his expert advice on computational optimization matters. Entries of a matrix E of size mxnx…xp, indexed by i,j,…,k, are denoted e ij…k. Conversely, given numbers , one can form a corresponding matrix of appropriate size. Z(P) will denote the optimal value of optimization problem (P).