Convergent Bounds for Stochastic Programs with Expected Value Constraints

Article

Abstract

This article describes a bounding approximation scheme for convex multistage stochastic programs (MSP) that constrain the conditional expectation of some decision-dependent random variables. Expected value constraints of this type are useful for modelling a decision maker’s risk preferences, but they may also arise as artifacts of stage-aggregation. We develop two finite-dimensional approximate problems that provide bounds on the (infinite-dimensional) original problem, and we show that the gap between the bounds can be made smaller than any prescribed tolerance. Moreover, the solutions of the approximate MSPs give rise to a feasible policy for the original MSP, and this policy’s optimality gap is shown to be smaller than the difference of the bounds. The considered problem class comprises models with integrated chance constraints and conditional value-at-risk constraints. No relatively complete recourse is assumed.

Keywords

Stochastic programming Approximation Bounds Expected value constraints Integrated chance constraints 

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References

  1. 1.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997) MATHGoogle Scholar
  2. 2.
    Kall, P., Wallace, S.W.: Stochastic Programming. Wiley, Chichester (1994) MATHGoogle Scholar
  3. 3.
    Prékopa, A.: Stochastic Programming. Kluwer Academic, Dordrecht (1995) Google Scholar
  4. 4.
    Wallace, S.W., Ziemba, W.T. (eds.): Applications of Stochastic Programming. MPS/SIAM Series on Optimization, vol. 5. SIAM, Philadelphia (2005) MATHGoogle Scholar
  5. 5.
    Kaut, M., Wallace, S.W.: Evaluation of scenario-generation methods for stochastic programming. Pac. J. Optim. 3(2), 257–271 (2007) MATHMathSciNetGoogle Scholar
  6. 6.
    Høyland, K., Wallace, S.W.: Generating scenario trees for multistage decision problems. Manag. Sci. 47(2), 295–307 (2001) CrossRefGoogle Scholar
  7. 7.
    Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett. 34, 1–8 (2006) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Shapiro, A.: Inference of statistical bounds for multistage stochastic programming problems. Math. Methods Oper. Res. 58(1), 57–68 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Koivu, M.: Variance reduction in sample approximations of stochastic programs. Math. Program. Ser. A 103, 463–485 (2005) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program. Ser. B 116(1), 461–479 (2009) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Pflug, G.Ch.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. Ser. B 89, 251–271 (2001) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hochreiter, R., Pflug, G.Ch.: Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Ann. Oper. Res. 156(1), 257–272 (2007) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Dupačová, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming: an approach using probability metrics. Math. Program. Ser. A 95, 493–511 (2003) MATHCrossRefGoogle Scholar
  14. 14.
    Heitsch, H., Römisch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24, 187–206 (2003) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rachev, S.T., Römisch, W.: Quantitative stability in stochastic programming: the method of probability metrics. Math. Oper. Res. 27, 792–818 (2002) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Dantzig, G.B., Infanger, G.: Large-scale stochastic linear programs–importance sampling and Benders decomposition. Comput. Appl. Math. I, 111–120 (1992) MathSciNetGoogle Scholar
  17. 17.
    Dempster, M.A.H., Thompson, R.T.: EVPI-based importance sampling solution procedures for multistage stochastic linear programmes on parallel MIMD architectures. Ann. Oper. Res. 90, 161–184 (1999) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ermoliev, Y.M., Gaivoronski, A.A.: Stochastic quasigradient methods for optimization of discrete event systems. Ann. Oper. Res. 39, 1–39 (1992) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Higle, J.L., Sen, S.: Stochastic decomposition: an algorithm for two-stage linear programs with recourse. Math. Oper. Res. 16, 650–669 (1991) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Infanger, G.: Planning under Uncertainty: Solving Large-Scale Stochastic Linear Programs. Boyd and Fraser, Danvers (1994) MATHGoogle Scholar
  21. 21.
    Birge, J.R., Wets, R.J.-B.: Computing bounds for stochastic programming problems by means of a generalized moment problem. Math. Oper. Res. 12, 149–162 (1987) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Dupačová, J.: On minimax solutions of stochastic linear programming problems. Časopis Pěstování Mat. 91, 423–429 (1966). (As Žáčková) Google Scholar
  23. 23.
    Frauendorfer, K.: Solving SLP recourse problems with arbitrary multivariate distributions—the dependent case. Math. Oper. Res. 13, 377–394 (1988) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Gassmann, G., Ziemba, W.T.: A tight upper bound for the expectation of a convex function of a multivariate random variable. Math. Program. Study 27, 39–53 (1986) MATHMathSciNetGoogle Scholar
  25. 25.
    Kall, P.: An upper bound for SLP using first and total second moments. Ann. Oper. Res. 30, 267–276 (1991) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Madansky, A.: Inequalities for stochastic linear programming problems. Manag. Sci. 6, 197–204 (1960) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Edirisinghe, N.C.P., Ziemba, W.T.: Bounding the expectation of a saddle function with application to stochastic programming. Math. Oper. Res. 19, 314–340 (1994) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Edirisinghe, N.C.P., Ziemba, W.T.: Bounds for two-stage stochastic programs with fixed recourse. Math. Oper. Res. 19, 292–313 (1994) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Frauendorfer, K.: Stochastic Two-Stage Programming. Lecture Notes in Economics and Mathematical Systems, vol. 392. Springer, Berlin (1992) MATHGoogle Scholar
  30. 30.
    Frauendorfer, K.: Multistage stochastic programming: error analysis for the convex case. Z. Oper. Res. 39(1), 93–122 (1994) MATHMathSciNetGoogle Scholar
  31. 31.
    Frauendorfer, K.: Barycentric scenario trees in convex multistage stochastic programming. Math. Program. 75(2), 277–294 (1996) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Kuhn, D.: Generalized Bounds for Convex Multistage Stochastic Programs. Lecture Notes in Economics and Mathematical Systems, vol. 548. Springer, Berlin (2004) Google Scholar
  33. 33.
    Wright, S.E.: Primal-dual aggregation and disaggregation for stochastic linear programs. Math. Oper. Res. 19(4), 893–908 (1994) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Kuhn, D.: Aggregation and discretization in multistage stochastic programming. Math. Program. Ser. A 113(1), 61–94 (2008) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Prékopa, A.: Contributions to the theory of stochastic programming. Math. Program. 4, 202–221 (1973) MATHCrossRefGoogle Scholar
  36. 36.
    Charnes, A., Cooper, W.W.: Chance-constrained programming. Manag. Sci. 6, 73–79 (1959/1960) CrossRefMathSciNetGoogle Scholar
  37. 37.
    Klein Haneveld, W.K.: Duality in Stochastic Linear and Dynamic Programming. Lecture Notes in Economics and Mathematical Systems, vol. 274. Springer, Berlin (1985) Google Scholar
  38. 38.
    Klein Haneveld, W.K., van der Vlerk, M.H.: Integrated chance constraints: reduced forms and an algorithm. Comput. Manag. Sci. 3(4), 245–269 (2006) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000) Google Scholar
  40. 40.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443–1471 (2002) CrossRefGoogle Scholar
  41. 41.
    Rockafellar, R.T., Wets, R.J.-B.: The optimal recourse problem in discrete time: L 1-multipliers for inequality constraints. SIAM J. Control Optim. 16, 16–36 (1978) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelphia (1974) MATHGoogle Scholar
  43. 43.
    Rockafellar, R.T., Wets, R.J.-B.: Nonanticipativity and L 1-martingales in stochastic optimization problems. Math. Program. Study 6, 170–187 (1976) MathSciNetGoogle Scholar
  44. 44.
    Rockafellar, R.T., Wets, R.J.-B.: Stochastic convex programming: basic duality. Pac. J. Math. 62, 173–195 (1976) MATHMathSciNetGoogle Scholar
  45. 45.
    Rockafellar, R.T., Wets, R.J.-B.: Stochastic convex programming: relatively complete recourse and induced feasibility. SIAM J. Control Optim. 14, 574–589 (1976) MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Rockafellar, R.T., Wets, R.J.-B.: Stochastic convex programming: singular multipliers and extended duality. Pac. J. Math. 62, 507–522 (1976) MATHMathSciNetGoogle Scholar
  47. 47.
    Dunford, N., Schwartz, J.T.: Linear Operators: Part I. Wiley, New York/Chichester/Brisbane/Toronto/Singapore (1988) MATHGoogle Scholar
  48. 48.
    Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30(1), 245–256 (2005) MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim. 17, 511–525 (2006) MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Casey, M.S., Sen, S.: The scenario generation algorithm for multistage stochastic linear programming. Math. Oper. Res. 30(3), 615–631 (2005) MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Kuhn, D., Parpas, P., Rustem, B.: Threshold accepting approach to improve bound-based approximations for portfolio optimization. In: Kontoghiorghes, E., Rustem, B., Winker, P. (eds.) Computational Methods in Financial Engineering, pp. 3–26. Springer, Berlin (2008) CrossRefGoogle Scholar
  52. 52.
    Kuhn, D., Parpas, P., Rustem, B.: Bound-based decision rules in multistage stochastic programming. Kybernetika 44(2), 134–150 (2008) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of ComputingImperial College of Science, Technology, and MedicineLondonUK

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