Mathematical Programming Computation

, Volume 10, Issue 4, pp 631–658 | Cite as

ADMM for the SDP relaxation of the QAP

  • Danilo Elias Oliveira
  • Henry Wolkowicz
  • Yangyang XuEmail author
Full Length Paper


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.


Quadratic assignment problem Semidefinite programming relaxation Alternating direction method of multipliers Facial reduction Doubly nonnegative Large scale 

Mathematics Subject Classification

90C22 90B80 90C46 90C06 90-08 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  • Danilo Elias Oliveira
    • 1
  • Henry Wolkowicz
    • 1
  • Yangyang Xu
    • 2
    Email author
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

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