Mathematical Programming

, Volume 114, Issue 2, pp 249–275 | Cite as

Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

  • Nilay Noyan
  • Andrzej Ruszczyński


Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming. However, problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem.


Stochastic programming Stochastic dominance Valid inequalities Disjunctive cuts Conditional value at risk 


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  1. 1.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Technical report MSRR 348, Carnegie Mellon University (1974)Google Scholar
  2. 2.
    Balas, E.: Disjunctive programming: cutting planes from logical conditions. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M.(eds.), Nonlinear Programming 2, Academics Press, New York, 279–312 (1975)Google Scholar
  3. 3.
    Balas E. (1979). Disjunctive programming. Ann Discr. Math. 5: 3–51 zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Balas E., Ceria S. and Cornuèjols G. (1993). A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Programm. 58: 295–324 CrossRefGoogle Scholar
  5. 5.
    Balas E., Ceria S. and Cornuèjols G. (1993). Solving mixed 0–1 programs by a lift-and-project method. SODA 1993: 232–242 Google Scholar
  6. 6.
    Balas E., Ceria S. and Cornuèjols G. (1996). Mixed 0–1 programming by lift-and-project in a branch-and-cut framework. Manage. Sci. 42: 1229–1246 zbMATHGoogle Scholar
  7. 7.
    Balas, E., Perregaard, M.: Generating cuts from multiple-tem disjunctions. In: Aardal, K., Gerards, B. (eds.) Proceedings of IPCO VIII. Lecture Notes in Computer Science 2081, 348–360 (2001)Google Scholar
  8. 8.
    Blair C.E. and Jeroslow R.G. (1978). A converse for disjunctive constraints. J. Optim. Theory Appl. 25: 195–206 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ceria, S., Soares, J.: Disjunctive cut generation for mixed 0–1 programs: duality and lifting. Working Paper, Graduate School of Business, Columbia University (1997)Google Scholar
  10. 10.
    Dentcheva D. and Ruszczyński A. (2003). Optimization with stochastic dominance constraints. SIAM J. Optim. 14: 548–566 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dentcheva D. and Ruszczyński A. (2004). Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints. Math. Programm. 99: 329–350 zbMATHCrossRefGoogle Scholar
  12. 12.
    Dentcheva D. and Ruszczyński A. (2004). Convexification of stochastic ordering. C. R. Acad. Bulgare Sci. 57(4): 7–14 Google Scholar
  13. 13.
    Dentcheva D. and Ruszczyński A. (2004). Semi-infinite probabilistic optimization: first order stochastic dominance constraints. Optimization 53: 583–601 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dentcheva D. and Ruszczyński A. (2006). Portfolio optimization with first order stochastic dominance constraints. J. Banking Financ 30(2): 433–451 CrossRefGoogle Scholar
  15. 15.
    Fishburn P.C. (1970). Utility Theory for Decision Making. Wiley, New York zbMATHGoogle Scholar
  16. 16.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modelling Language for Mathematical Programming. The Scientific Press (1993)Google Scholar
  17. 17.
    Hadar J. and Russell W. (1969). Rules for ordering uncertain prospects. Am. Econ. Rev. 59: 25–34 Google Scholar
  18. 18.
    Hanoch G. and Levy H. (1969). The efficiency analysis of choices involving risk. Rev. Econ. Stud. 36: 335–346 zbMATHCrossRefGoogle Scholar
  19. 19.
    Klatte D. and Henrion R. (1998). Regularity and stability in nonlinear semi-infinite optimization. Nonconvex optim. Appl. 25: 69–102 MathSciNetGoogle Scholar
  20. 20.
    ILOG CPLEX: CPLEX 9.0 Users Manual and Reference Manual, ILOG CPLEX Division, Incline Village, NV (2005)Google Scholar
  21. 21.
    Klein Haneveld, W.K.: Duality in Stochastic Linear and Dynamic Programming. Lecture Notes in Economics and Mathematical Systems, Vol. 274. Springer, New York (1986)Google Scholar
  22. 22.
    Klein Haneveld W.K. and VanDer Vlerk M.H. (2006). Integrated chance constraints: reduced forms and an algorithm. Comput. Manage. Sci. 3(4): 245–269 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lehmann E. (1955). Ordered families of distributions. Ann. Math. Stat. 26: 399–419 CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Levy H. (1992). Stochastic dominance and expected utility: survey and analysis. Manage Sci. 38: 555–593 zbMATHCrossRefGoogle Scholar
  25. 25.
    Mann H.B. and Whitney D.R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Stat. 18: 50–60 CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Markowitz H.M. (1952). Portfolio Selection. J. Finan. 7: 77–91 CrossRefGoogle Scholar
  27. 27.
    Markowitz H.M. (1959). Portfolio Selection. Wiley, New York Google Scholar
  28. 28.
    Noyan N., Rudolf G. and Ruszczyński A. (2005). Relaxations of linear programming problems with first order stochastic dominance constraints. Oper. Res. Lett. 103: 784–797 Google Scholar
  29. 29.
    Noyan, N.: Optimization with first order stochastic dominance constraint. Doctoral Dissertation, Graduate School of New Brunswick, Rutgers, The State University of New Jersey, Piscataway, USA (2006)Google Scholar
  30. 30.
    Ogryczak W. and Ruszczyński A. (1999). From stochastic dominance to mean-risk models: semideviations as risk measures. Eur. J. Oper. Res. 116: 33–50 zbMATHCrossRefGoogle Scholar
  31. 31.
    Ogryczak W. and Ruszczyński A. (2001). Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13: 60–78 CrossRefGoogle Scholar
  32. 31.
    Perregaard, M.: Generating disjunctive cuts for mixed integer programs. Doctoral Dissertation, Graduate School of Industrial Administration, Schenley Park, Pittsburgh, Carnegie Mellon University (2003)Google Scholar
  33. 33.
    Quirk J.P. and Saposnik R. (1962). Admissibility and measurable utility functions. Rev. Econ. Stud. 29: 140–146 CrossRefGoogle Scholar
  34. 34.
    Rockafellar R.T. and Uryasev S. (2000). Optimization of conditional value at risk. J. Risk 2: 21–41 Google Scholar
  35. 35.
    Rothschild M. and Stiglitz J.E. (1969). Increasing risk: I. A definition. J. Econom. Theory 2: 225–243 CrossRefMathSciNetGoogle Scholar
  36. 36.
    Rudolf, G., Ruszczyński, A.: A dual approach to linear stochastic optimization problems with second order dominance constraints (in preparation) (2006)Google Scholar
  37. 37.
    Ruszczyński A. and Vanderbei R.J. (2003). Frontiers of stochastically nondominated portfolios. Econometrica 71: 1287–1297 CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Whitmore G.A. and Findlay M.C. (1978). Stochastic dominance: an approach to decision-making under risk. D.C.Heath, Lexington Google Scholar

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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Manufacturing Systems and Industrial Engineering Program, Faculty of Engineering and Natural SciencesSABANCI UniversityOrhanli, Tuzla, IstanbulTurkey
  2. 2.Department of Management Science and Information SystemsRutgers UniversityPiscatawayUSA

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