Abstract
In this paper, we extend the purely dual formulation that we recently proposed for the three-dimensional assignment problems to solve the more general multidimensional assignment problem. The convex dual representation is derived and its relationship to the Lagrangian relaxation method that is usually used to solve multidimensional assignment problems is investigated. Also, we discuss the condition under which the duality gap is zero. It is also pointed out that the process of Lagrangian relaxation is essentially equivalent to one of relaxing the binary constraint condition, thus necessitating the auction search operation to recover the binary constraint. Furthermore, a numerical algorithm based on the dual formulation along with a local search strategy is presented. The simulation results show that the proposed algorithm outperforms the traditional Lagrangian relaxation approach in terms of both accuracy and computational efficiency.
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Acknowledgements
(1) We would like to thank the two anonymous reviewers for their comments and suggestions to improve the paper. (2) The paper was originally taken from the PhD thesis work of the first author, and later was revised after he lost his job in the pandemic and supported by the Canada Emergency Response Benefit (CERB). It would be impossible to complete this work without the CERB support to which the first author is deeply grateful.
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Jingqun Li: Currently unemployed: looking for a machine learning position for conducting research on combinatorial optimization, computer vision and natural language processing.
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Li, J., Kirubarajan, T., Tharmarasa, R. et al. A dual approach to multi-dimensional assignment problems. J Glob Optim 81, 691–716 (2021). https://doi.org/10.1007/s10898-020-00988-8
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DOI: https://doi.org/10.1007/s10898-020-00988-8