Biq Mac

Solving Max-cut to optimality by intersecting semidefinite and polyhedral relaxations. We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem. The experiments show that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming-based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to $n = 100$, independent of the density. For some problems of special structure, we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so.

References in zbMATH (referenced in 46 articles , 1 standard article )

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  1. Chen, Wei-An; Zhu, Zhen; Kong, Nan: A Lagrangian decomposition approach to computing feasible solutions for quadratic binary programs (2018)
  2. Gally, Tristan; Pfetsch, Marc E.; Ulbrich, Stefan: A framework for solving mixed-integer semidefinite programs (2018)
  3. Fampa, Marcia; Lee, Jon; Melo, Wendel: On global optimization with indefinite quadratics (2017)
  4. Fampa, Marcia; Pimentel, Wagner: Linear programing relaxations for a strategic pricing problem in electricity markets (2017)
  5. Lima, Ricardo M.; Grossmann, Ignacio E.: On the solution of nonconvex cardinality Boolean quadratic programming problems: a computational study (2017)
  6. Ben-Ameur, Walid; Glorieux, Antoine; Neto, José: From graph orientation to the unweighted maximum cut (2016)
  7. Bergman, David; Cire, Andre A.; van Hoeve, Willem-Jan; Hooker, J.N.: Discrete optimization with decision diagrams (2016)
  8. Boukouvala, Fani; Misener, Ruth; Floudas, Christodoulos A.: Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO (2016)
  9. Dong, Hongbo: Relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations (2016)
  10. John, Maximilian; Karrenbauer, Andreas: A novel SDP relaxation for the quadratic assignment problem using cut pseudo bases (2016)
  11. Kim, Sunyoung; Kojima, Masakazu; Toh, Kim-Chuan: A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems (2016)
  12. Létocart, Lucas; Wiegele, Angelika: Exact solution methods for the $k$-item quadratic knapsack problem (2016)
  13. Rendl, F.: Semidefinite relaxations for partitioning, assignment and ordering problems (2016)
  14. van Dam, E.R.; Sotirov, R.: New bounds for the $\max$-$k$-cut and chromatic number of a graph (2016)
  15. Delling, Daniel; Fleischman, Daniel; Goldberg, Andrew V.; Razenshteyn, Ilya; Werneck, Renato F.: An exact combinatorial algorithm for minimum graph bisection (2015)
  16. Goldberg, Noam; Leyffer, Sven: An active-set method for second-order conic-constrained quadratic programming (2015)
  17. Hungerländer, P.: A semidefinite optimization approach to the target visitation problem (2015)
  18. Lieder, Felix; Rad, Fatemeh Bani Asadi; Jarre, Florian: Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques (2015)
  19. Xia, Yong; Xing, Wenxun: Parametric Lagrangian dual for the binary quadratic programming problem (2015)
  20. Allouche, David; André, Isabelle; Barbe, Sophie; Davies, Jessica; de Givry, Simon; Katsirelos, George; O’Sullivan, Barry; Prestwich, Steve; Schiex, Thomas; Traoré, Seydou: Computational protein design as an optimization problem (2014)

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