Mathematical Programming

, Volume 106, Issue 2, pp 225–236 | Cite as

Perspective cuts for a class of convex 0–1 mixed integer programs

Article

Abstract

We show that the convex envelope of the objective function of Mixed-Integer Programming problems with a specific structure is the perspective function of the continuous part of the objective function. Using a characterization of the subdifferential of the perspective function, we derive “perspective cuts”, a family of valid inequalities for the problem. Perspective cuts can be shown to belong to the general family of disjunctive cuts, but they do not require the solution of a potentially costly nonlinear programming problem to be separated. Using perspective cuts substantially improves the performance of Branch & Cut approaches for at least two models that, either “naturally” or after a proper reformulation, have the required structure: the Unit Commitment problem in electrical power production and the Mean-Variance problem in portfolio optimization.

Keywords

Mixed-Integer Programs Valid Inequalities Unit Commitment problem Portfolio Optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahn, S., Escudero, L.F., Guignard-Spielberg, M.: On modeling robust policies for financial trading. In: T.A. Ciriani and R.L. Leachman (eds.) Optimization in Industry 2, Wiley Chichester, 1994 pp 163–184Google Scholar
  2. 2.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Mathematical Programming 58, 295–324 (1993)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Borghetti, A., Frangioni, A., Lacalandra, F., Nucci, C.A.: Lagrangian heuristics based on disaggregated bundle methods for hydrothermal unit commitment. IEEE Transactions on Power Systems 18 (1), 1–10 (2003)Google Scholar
  4. 4.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Mathematical Programming 86, 595–614 (1999)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Duran, M.A., Grossmann, I.E: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming 36, 307–339 (1986)MATHMathSciNetGoogle Scholar
  6. 6.
    Frangioni, A., Gentile, C.: Perspective cuts for 0–1 mixed integer programs. Technical report 577, Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” (IASI-CNR), 2002Google Scholar
  7. 7.
    Frangioni, A.: Generalized bundle methods. SIAM Journal on Optimization 13 (1), 117–156 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Frangioni, A.: About lagrangian methods in integer optimization. Annals of Operations Research, To appear 2005Google Scholar
  9. 9.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex analysis and minimization algorithms I–- fundamentals. Grundlehren Math. Wiss. 305 Springer-Verlag, New York 1993Google Scholar
  10. 10.
    Jobst, N.J., Horniman, M.D., Lucas, C.A., Mitra, G.: Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative Finance 1, Wiley Chichester, 2001, pp 1–13Google Scholar
  11. 11.
    Kallrath, J., Wilson, J.M.: Business optimization. Macmillan Press Ltd. Houndmills, 1997Google Scholar
  12. 12.
    Markowitz, H.M.: Portfolio selection. Journal of Finance 7, 77–91 (1952)CrossRefGoogle Scholar
  13. 13.
    Padberg, M.W., Rinaldi, G.: A branch and cut algorithm for resolution of large scale symmetric salesman problems. SIAM Review 33, 60–100 (1991)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Pardalos, P.M., Rodgers, G.P.: Computing aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45, 131–144 (1990)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rikun, A.D.: A convex envelope formula for multilinear functions. Journal of Global Optimimization 10, 425–437 (1997)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0-1 mixed convex programming. Mathematical Programming 86, 515–532 (1999)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions and envelopes of lower semi-continuous functions. Mathematical Programming 93, 515–532 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zamora, J.M., Grossmann, I.E.: A global MINLP optimization algorithm for the synthesis of heat exchanger networks with no stream splits. Comput & Chem. Engin. 22, 367–384 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of PisaPisaItaly
  2. 2.Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, C.N.R.RomeItaly

Personalised recommendations