A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem

The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpens...

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Bibliographic Details
Published inMathematics of operations research Vol. 34; no. 4; pp. 1008 - 1022
Main Authors Ding, Yichuan, Wolkowicz, Henry
Format Journal Article
LanguageEnglish
Published Linthicum INFORMS 01.11.2009
Institute for Operations Research and the Management Sciences
Subjects
Analysis
Assignment problem
Branch & bound algorithms
Comparative analysis
Eigenvalues
interior point methods
Interior points
large-scale problems
Mathematical inequalities
Mathematical permutation
Mathematical theorems
Mathematical vectors
Matrices
Objective functions
quadratic assignment problem
Quadratic programming
Relaxation methods (Mathematics)
semidefinite programming relaxations
Studies
United States
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Summary:The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpensive bounds. In this paper, we introduce a new semidefinite programming (SDP) relaxation for generating bounds for the QAP in the trace formulation. We apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function. This exploits the matrix structure of QAP and results in a relaxation with much smaller dimension than other current SDP relaxations. We compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation (QPB). We find that our method provides stronger bounds on average and is adaptable for branch-and-bound methods.
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ISSN:0364-765X
1526-5471
DOI:10.1287/moor.1090.0419