Mathematical Programming Computation

, Volume 5, Issue 1, pp 75–112 | Cite as

Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints

  • I. R. de FariasJr.Email author
  • E. Kozyreff
  • R. Gupta
  • M. Zhao
Full Length Paper


We report and analyze the results of our computational testing of branch-and-cut for piecewise linear optimization using the cutting planes given recently by Zhao and de Farias. Besides evaluating the performance of the cuts, we evaluate the effect of formulation on the performance of branch-and-cut. Finally, we report and analyze results on piecewise linear optimization problems with semi-continuous constraints.


Piecewise linear optimization Mixed-integer programming  Knapsack problem Special ordered set Semi-continuous variable  Polyhedral method Branch-and-cut 

Mathematics Subject Classification




This research was partially supported by the Office of Naval Research and the National Science Foundation through grants N000140910332 and CMMI-0620755, respectively. Their support is gratefully acknowledged. We are grateful to George Nemhauser and Juan-Pablo Vielma for making available to us the instances of their paper [30]. We are also grateful to Zhonghao Gu and Ed Rothberg for enlightening discussions. Finally, we are grateful to the anonymous referees and the editors, for several valuable suggestions.


  1. 1.
    Beale, E.M.L.: Two transportation problems. In: Kreweras, G., Morlat, G. (eds.) Proceedings of the Third International Conference on Operational Research, Dunod, pp. 780–788 (1963)Google Scholar
  2. 2.
    Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for nonconvex problems using ordered sets of variables. In: Lawrence, J. (ed.) Proceedings of the Fifth International Conference on Operations Research, Tavistock Publications, pp. 447–454 (1970)Google Scholar
  3. 3.
    Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Crowder, H., Johnson, E.L., Padberg, M.: Solving large-scale zero-one linear programming problems. Oper. Res. 31, 803–834 (1983)zbMATHCrossRefGoogle Scholar
  5. 5.
    Croxton, K.L., Gendron, B., Magnanti, T.L.: Models and methods for merge-in-transit operations. Transp. Sci. 37, 1–22 (2003)CrossRefGoogle Scholar
  6. 6.
    Croxton, K.L., Gendron, B., Magnanti, T.L.: Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs. Operations Research Center, Massachusetts Institute of Technology, Cambridge (2003)Google Scholar
  7. 7.
    Dantzig, G.B.: On the significance of solving linear programming problems with some integer variables. Econometrica 28, 30–44 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    de Farias, I.R. Jr.: Semi-continuous cuts for mixed-integer programming. In: Bienstock, D., Nemhauser, G.L. (eds.) Integer Programming and Combinatorial Optimization (IPCO). Lecture Notes in Computer Science, vol. 3064, pp. 163–177, Springer (2004)Google Scholar
  9. 9.
    de Farias, I.R. Jr., Johnson, E.L., Nemhauser, G.L.: A generalized assignment problem with special ordered sets: a polyhedral approach. Math. Program. 89, 187–203 (2000)Google Scholar
  10. 10.
    de Farias, I.R Jr., Johnson, E.L., Nemhauser, G.L.: Branch-and-cut for combinatorial optimization problems without auxiliary binary variables. Knowl. Eng. Rev. 16, 25–39 (2001)Google Scholar
  11. 11.
    de Farias, I.R. Jr., Kozyreff, E., Gupta, R., Zhao, M.: Branch-and-Cut for Separable Piecewise Linear Optimization and Intersection with Semi-Continuous Constraints. Texas Tech University, USA (2011)Google Scholar
  12. 12.
    de Farias, I.R. Jr., Nemhauser, G.L.: A polyhedral study of the cardinality constrained knapsack problem. Math. Program. 96, 439–467 (2003)Google Scholar
  13. 13.
    de Farias, I.R. Jr., Zhao, M.: A polyhedral study of the semi-continuous knapsack problem. Math. Programm. (2011, submitted)Google Scholar
  14. 14.
    de Farias, I.R. Jr., Zhao, M., Zhao, H.: A special ordered set approach for optimizing a discontinuous separable piecewise linear function. Oper. Res. Lett. 36, 234–238 (2008)Google Scholar
  15. 15.
    Fourer, R., Gay, D.M., Kerninghan, B.W.: AMPL: A Modeling Language for Mathematical Programming. The Scientific Press, USA (1993)Google Scholar
  16. 16.
    Gu, Z.: Personal communicationGoogle Scholar
  17. 17.
  18. 18.
    Keha, A.B., de Farias, I.R. Jr., Nemhauser, G.L.: Models for representing piecewise linear cost functions. Oper. Res. Lett. 32, 44–48 (2004)Google Scholar
  19. 19.
    Keha, A.B., de Farias, I.R. Jr., Nemhauser, G.L.: A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54, 847–858 (2006)Google Scholar
  20. 20.
    Konno, H., Wijayanayake, A.: Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Math. Program. 89, 233–250 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Markowitz, H.M., Manne, A.S.: On the solution of discrete programming problems. Econometrica 25, 84–110 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Martin, A., Möller, M., Moritz, S.: Mixed-integer models for the stationary case of gas network optimization. Math. Program. 105, 563–582 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)zbMATHCrossRefGoogle Scholar
  24. 24.
    Perold, A.F.: Large-scale portfolio optimization. Manag. Sci. 30, 1143–1160 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Sioshansi, R., O’Neill, R.O., Oren, S.S.: Economic consequences of alternative solution methods for centralized unit commitment in day-ahead electricity markets. IEEE Trans. Power Syst. 23, 344–352 (2008)CrossRefGoogle Scholar
  26. 26.
    Takriti, S., Birge, J.R., Long, E.: A stochastic model for the unit commitment problem. IEEE Trans. Power Syst. 11, 1497–1508 (1996)CrossRefGoogle Scholar
  27. 27.
    Takriti, S., Krasenbrink, B., Wu, L.S.Y.: Incorporating fuel constraints and electricity spot prices into the stochastic unit commitment problem. Oper. Res. 48, 268–280 (2000)CrossRefGoogle Scholar
  28. 28.
    Tomlin, J.A.: Special ordered sets and an application to gas supply operations planning. Math. Program. 42, 69–84 (1988)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Vielma, J.P., Ahmed, S., Nemhauser, G.L.: Mixed-integer models for nonseparable piecewise linear optimization: unifying framework and extensions. Oper. Res. 58, 303–315 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128, 49–72 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Zhang, M., Guan, Y.: Two-Stage Robust Unit Commitment Problem. University of Florida, USA (2009)Google Scholar
  32. 32.
    Zhao, M., de Farias, I.R. Jr.: The Piecewise Linear Optimization Polytope: New Inequalities and Intersection with Semi-Continuous Constraints. Math. Program. (2012, in press)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  • I. R. de FariasJr.
    • 1
    Email author
  • E. Kozyreff
    • 1
  • R. Gupta
    • 1
  • M. Zhao
    • 2
  1. 1.Department of Industrial EngineeringTexas Tech UniversityLubbockUSA
  2. 2.SASCaryUSA

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