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ADMM for the SDP relaxation of the QAP

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Abstract

Semidefinite programming, SDP, relaxations have proven to be extremely strong for many hard discrete optimization problems. This is in particular true for the quadratic assignment problem, QAP, arguably one of the hardest NP-hard discrete optimization problems. There are several difficulties that arise in efficiently solving the SDP relaxation, e.g., increased dimension; inefficiency of the current primal–dual interior point solvers in terms of both time and accuracy; and difficulty and high expense in adding cutting plane constraints. We propose using the alternating direction method of multipliers ADMM in combination with facial reduction, FR, to solve the SDP relaxation. This first order approach allows for: inexpensive iterations, a method of cheaply obtaining low rank solutions; and a trivial way of exploiting the FR for adding cutting plane inequalities. In fact, we solve the doubly nonnegative, DNN, relaxation that includes both the SDP and all the nonnegativity constraints. When compared to current approaches and current best available bounds we obtain robustness, efficiency and improved bounds.

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Notes

  1. We do not include the times as they were much greater than those by the ADMM approach, e.g., hours instead of minutes and a day instead of an hour.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    Danilo Elias Oliveira & Henry Wolkowicz

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, USA

    Yangyang Xu

Authors
  1. Danilo Elias Oliveira
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  2. Henry Wolkowicz
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  3. Yangyang Xu
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Corresponding author

Correspondence to Yangyang Xu.

Additional information

The code can be downloaded from the author’s webpage https://xu-yangyang.github.io/ADMM_QAP/ The software that was reviewed as part of this submission was given the DOI (Digital Object Identifier) 10.5281/zenodo.1412139.

This work is partially supported by NSERC and AFOSR. The first version of this paper appeared in optimization online, Dec. 16, 2015 and in arXiv:1512.05448, Dec. 17, 2015

Research supported by The Natural Sciences and Engineering Research Council of Canada and by AFOSR.

Research partly supported by NSF Grant DMS-1719549.

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Oliveira, D.E., Wolkowicz, H. & Xu, Y. ADMM for the SDP relaxation of the QAP. Math. Prog. Comp. 10, 631–658 (2018). https://doi.org/10.1007/s12532-018-0148-3

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  • DOI: https://doi.org/10.1007/s12532-018-0148-3

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