Mathematical Programming

, Volume 150, Issue 1, pp 99–129 | Cite as

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

  • James Ostrowski
  • Miguel F. AnjosEmail author
  • Anthony Vannelli
Full Length Paper Series B


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.


Symmetry Integer programming Orbital branching  Orbitopes  Unit commitment 

Mathematics Subject Classification

90C10 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. 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
  2. 2.
    Arroyo, J.M., Conejo, A.J.: Optimal response of a thermal unit to an electricity spot market. IEEE Trans. Power Syst. 15(3), 1098–1104 (2000)CrossRefGoogle Scholar
  3. 3.
    Carrion, M., Arroyo, J.: A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Trans. Power Syst. 21(3), 1371–1378 (2006)CrossRefGoogle Scholar
  4. 4.
    Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106, 226–236 (2006)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Frangioni, A., Gentile, C., Lacalandra, F.: Tighter approximated milp formulations for unit commitment problems. IEEE Trans. Power Syst. 24(1), 105–113 (2009)CrossRefGoogle Scholar
  6. 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
  7. 7.
    Gattermann, K., Parrilo, P.: Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192, 95–128 (2004)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Jeroslow, R.: Trivial integer programs unsolvable by branch-and-bound. Math. Program. 6, 105–109 (1974)CrossRefzbMATHGoogle Scholar
  9. 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
  10. 10.
    Kaibel, V., Peinhardt, M., Pfetsch, M.E.: Orbitopal fixing. Discret. Optim. 8(4), 595–610 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kaibel, V., Pfetsch, M.: Packing and partitioning orbitopes. Math. Program. 114, 1–36 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Liu, C., Shahidehpour, M., Li, Z., Fotuhi-Firuzabad, M.: Component and mode models for the short-term scheduling of combined-cycle units. IEEE Trans. Power Syst. 24(2), 976–990 (2009)CrossRefGoogle Scholar
  13. 13.
    Margot, F.: Pruning by isomorphism in branch-and-cut. Math. Program. 94, 71–90 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Margot, F.: Exploiting orbits in symmetric ILP. Math. Program. Ser. B 98, 3–21 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 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
  16. 16.
    Ostrowski, J., Anjos, M.F., Vannelli, A.: Tight mixed integer linear programming formulations for the unit commitment problem. IEEE Trans. Power Syst. 27(1), 39–46 (2012)CrossRefGoogle Scholar
  17. 17.
    Ostrowski, J., Linderoth, J., Rossi, F., Smriglio, S.: Orbital branching. Math. Program. 126(1), 147–178 (2009)CrossRefMathSciNetGoogle Scholar
  18. 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
  19. 19.
    Sherali, H.D., Smith, J.C.: Improving zero-one model representations via symmetry considerations. Manag. Sci. 47(10), 1396–1407 (2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    Zhai, Q., Guan, X., Cui, J.: Unit commitment with identical units successive subproblem solving method based on Lagrangian relaxation. IEEE Trans. Power Syst. 17(4), 1250–1257 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • James Ostrowski
    • 1
  • Miguel F. Anjos
    • 2
    Email author
  • Anthony Vannelli
    • 3
  1. 1.Department of Industrial and Systems EngineeringUniversity of Tennessee KnoxvilleKnoxvilleUSA
  2. 2.Canada Research Chair in Discrete Nonlinear Optimization in EngineeringGERAD and École Polytechnique de MontréalMontrealCanada
  3. 3.School of EngineeringUniversity of GuelphGuelphCanada

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