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A proximal DC approach for quadratic assignment problem

Abstract

In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method. We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem under some suitable assumptions. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time.

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Acknowledgements

We would like to thank Dr. Xudong Li and Dr. Ying Cui for many helpful discussions on this work. Also, we would like to thank the two anonymous referees and the associate editor for their valuable comments and constructive suggestions which have helped to improve the quality of this paper.

Funding

X. Zhao: The research of this author was supported by the National Natural Science Foundation of China under projects No. 11871002 and the General Program of Science and Technology of Beijing Municipal Education Commission No. KM201810005004. C. Ding: The research of this author was supported by the National Natural Science Foundation of China under projects No. 11671387, No. 11531014 and No. 11688101 and the Beijing Natural Science Foundation (Z190002).

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Correspondence to Chao Ding.

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Jiang, Z., Zhao, X. & Ding, C. A proximal DC approach for quadratic assignment problem. Comput Optim Appl 78, 825–851 (2021). https://doi.org/10.1007/s10589-020-00252-5

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Keywords

  • Quadratic assignment problem
  • Doubly nonnegative programming
  • Augmented Lagrangian method
  • Rank constraint

Mathematics Subject Classification

  • 90C22
  • 90C25
  • 90C26
  • 90C27