Modified orbital branching for structured symmetry with an application to unit commitment
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The past decade has seen advances in general methods for symmetry breaking in mixed-integer linear programming. These methods are advantageous for general problems with general symmetry groups. Some important classes of mixed integer linear programming problems, such as bin packing and graph coloring, contain highly structured symmetry groups. This observation has motivated the development of problem-specific techniques. In this paper we show how to strengthen orbital branching in order to exploit special structures in a problem’s symmetry group. The proposed technique, to which we refer as modified orbital branching, is able to solve problems with structured symmetry groups more efficiently. One class of problems for which this technique is effective is when the solution variables can be expressed as 0/1 matrices where the problem’s symmetry group contains all permutations of the columns. We use the unit commitment problem, an important problem in power systems, to demonstrate the strength of modified orbital branching.
KeywordsSymmetry Integer programming Orbital branching Orbitopes Unit commitment
Mathematics Subject Classification90C10 90C57 90C90
The authors thank an anonymous referee for the constructive reports that helped greatly improve this paper. The research of the first author was supported by NSF CMMI Grant 1332662. The research of the second and third authors was partially supported by NSERC, the Natural Sciences and Engineering Research Council of Canada.
- 1.Anjos, M.F.: Recent progress in modeling unit commitment problems. In: Zuluaga, L., Terlaky, T. (eds.) Modeling and Optimization: Theory and Applications: Selected Contributions from the MOPTA 2012 Conference, Springer Proceedings in Mathematics & Statistics, vol. 84. Springer (2013)Google Scholar
- 6.Friedman, E.J.: Fundamental domains for integer programs with symmetries. In: Dress, A.W.M., Xu, Y., Zhu, B. (eds.) Combinatorial Optimization and Applications, Lecture Notes in Computer Science, vol. 4616. Springer, pp. 146–153 (2007)Google Scholar
- 9.Kaibel, V., Loos, A.: Branched polyhedral systems. In: IPCO 2010: The Fourteenth Conference on Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 6080. Springer, pp. 177–190 (2010)Google Scholar
- 15.Margot, F.: Symmetry in integer linear programming. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 647–686. Springer, Berlin (2010)Google Scholar
- 18.Rajan, D., Takriti, S.: Minimum up/down polytopes of the unit commitment problem with start-up costs. Tech. rep., IBM Research Report (2005)Google Scholar