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

Author

Listed:
  • Naomi Graham

    (Department of Computer Science, The University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada)

  • Hao Hu

    (School of Mathematical and Statistical Sciences, Clemson University, Clemson, South Carolina 29634)

  • Jiyoung Im

    (Department of Combinatorics and Optimization, University of Waterloo, Waterloo, N2L 3G1 Ontario, Canada)

  • Xinxin Li

    (School of Mathematics, Jilin University, Changchun, Jilin 130012, China)

  • Henry Wolkowicz

    (Department of Combinatorics and Optimization, University of Waterloo, Waterloo, N2L 3G1 Ontario, Canada)

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.

Suggested Citation

  • Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.
  • Handle: RePEc:inm:orijoc:v:34:y:2022:i:4:p:2125-2143
    DOI: 10.1287/ijoc.2022.1161
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    References listed on IDEAS

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    1. Deren Han & Defeng Sun & Liwei Zhang, 2018. "Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 622-637, May.
    2. Xinxin Li & Ting Kei Pong & Hao Sun & Henry Wolkowicz, 2021. "A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem," Computational Optimization and Applications, Springer, vol. 78(3), pages 853-891, April.
    3. Rendl, F. & Sotirov, R., 2007. "Bounds for the quadratic assignment problem using the bundle method," Other publications TiSEM b6d298bc-77c9-4a6d-a043-5, Tilburg University, School of Economics and Management.
    4. Alexander Barvinok & Tamon Stephen, 2003. "The Distribution of Values in the Quadratic Assignment Problem," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 64-91, February.
    5. José F. S. Bravo Ferreira & Yuehaw Khoo & Amit Singer, 2018. "Semidefinite programming approach for the quadratic assignment problem with a sparse graph," Computational Optimization and Applications, Springer, vol. 69(3), pages 677-712, April.
    6. Burkard, R. E. & Karisch, S. & Rendl, F., 1991. "QAPLIB-A quadratic assignment problem library," European Journal of Operational Research, Elsevier, vol. 55(1), pages 115-119, November.
    7. Damek Davis & Wotao Yin, 2017. "Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 783-805, August.
    8. A. M. Geoffrion & G. W. Graves, 1976. "Scheduling Parallel Production Lines with Changeover Costs: Practical Application of a Quadratic Assignment/ LP Approach," Operations Research, INFORMS, vol. 24(4), pages 595-610, August.
    9. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
    10. Zvi Drezner, 2003. "A New Genetic Algorithm for the Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 15(3), pages 320-330, August.
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    1. Lennart Sinjorgo & Renata Sotirov, 2024. "On Solving MAX-SAT Using Sum of Squares," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 417-433, March.

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