Speeding up IP-based algorithms for constrained quadratic 0–1 optimization
- 183 Downloads
In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP). Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum-cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastic speedup in the solution of constrained quadratic 0–1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.
KeywordsQuadratic programming Maximum-cut problem Local cuts Quadratic knapsack Crossing minimization Similar subgraphs
Mathematics Subject Classification (2000)90C20 90C27 90C57
Unable to display preview. Download preview PDF.
- 3.Applegate, A., Bixby, R., Chvátal, V., Cook, W.: TSP cuts which do not conform to the template paradigm. In: Computational Combinatorial Optimization: Optimal or Provably Near-Optimal Solutions, Lecture Notes in Computer Science, vol. 2241, pp. 261–304. Springer (2001)Google Scholar
- 5.Buchheim, C., Liers, F., Oswald, M.: A basic toolbox for constrained quadratic 0/1 optimization. In: McGeoch, C.C. (ed.) WEA 2008: Workshop on Experimental Algorithms, Lecture Notes in Computer Science, vol. 5038, pp. 249–262. Springer (2008)Google Scholar
- 13.Deza M., Laurent M.: Geometry of Cuts and Metrics, Algorithms and Combinatorics, vol. 15. Springer, Berlin (1997)Google Scholar
- 17.Johnson, T.A.: New Linear-Programming based Solution Procedures for the Quadratic Assignment Problem. Ph.D. thesis, Graduate School of Clemson University (1992)Google Scholar
- 18.Liers F., Jünger M., Reinelt G., Rinaldi G.: Computing Exact Ground States of Hard Ising Spin Glass Problems by Branch-and-Cut. New Optimization Algorithms in Physics, pp. 47–68. Wiley, Weinheim (2004)Google Scholar
- 19.Rendl, F., Rinaldi, G., Wiegele, A.: A branch and bound algorithm for Max-Cut based on combining semidefinite and polyhedral relaxations. In: Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 4513, pp. 295–309. Springer, Berlin (2007)Google Scholar