ADMM for the SDP relaxation of the QAP. 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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References in zbMATH (referenced in 9 articles , 1 standard article )
Showing results 1 to 9 of 9.
- de Meijer, Frank; Sotirov, Renata: SDP-based bounds for the quadratic cycle cover problem via cutting-plane augmented Lagrangian methods and reinforcement learning (2021)
- Im, Jiyoung; Wolkowicz, Henry: A strengthened Barvinok-Pataki bound on SDP rank (2021)
- Li, Xinxin; Pong, Ting Kei; Sun, Hao; Wolkowicz, Henry: A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem (2021)
- Rendl, Franz; Sotirov, Renata; Truden, Christian: Lower bounds for the bandwidth problem (2021)
- Hu, Hao; Sotirov, Renata: On solving the quadratic shortest path problem (2020)
- Fang, Ethan X.; Liu, Han; Toh, Kim-Chuan; Zhou, Wen-Xin: Max-norm optimization for robust matrix recovery (2018)
- Ferreira, José F. S. Bravo; Khoo, Yuehaw; Singer, Amit: Semidefinite programming approach for the quadratic assignment problem with a sparse graph (2018)
- Oliveira, Danilo Elias; Wolkowicz, Henry; Xu, Yangyang: ADMM for the SDP relaxation of the QAP (2018)
- Ma, ShiQian; Yang, JunFeng: Applications of gauge duality in robust principal component analysis and semidefinite programming (2016)