Modified orbital branching for structured symmetry with an application to unit commitment

Abstract

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.

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Acknowledgments

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.

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Correspondence to Miguel F. Anjos.

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Ostrowski, J., Anjos, M.F. & Vannelli, A. Modified orbital branching for structured symmetry with an application to unit commitment. Math. Program. 150, 99–129 (2015). https://doi.org/10.1007/s10107-014-0812-y

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Keywords

  • Symmetry
  • Integer programming
  • Orbital branching
  • Orbitopes
  • Unit commitment

Mathematics Subject Classification

  • 90C10
  • 90C57
  • 90C90