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Automation and Remote Control

, Volume 80, Issue 6, pp 1041–1057 | Cite as

General Properties of Two-Stage Stochastic Programming Problems with Probabilistic Criteria

  • S. V. IvanovEmail author
  • A. I. KibzunEmail author
Stochastic Systems
  • 5 Downloads

Abstract

Two-stage stochastic programming problems with the probabilistic and quantile criteria in the general statement are considered. Sufficient conditions for the measurability of the loss function and also for the semicontinuity of the criterion functions are given. Sufficient conditions for the existence of optimal strategies are established. The equivalence of the a priori and a posteriori statements of the problems under study is proved. The application of the confidence method, which consists in the transition to a deterministic minimax problem, is described and justified. Sample approximations of the problems are constructed and also conditions under which the optimal strategies in the approximating problems converge to the optimal strategy in the original problem are presented. The results are illustrated by an example of the linear two-step problem. The two-stage problem with the probabilistic criterion is reduced to a mixed-integer problem.

Keywords

stochastic programming two-stage problem probabilistic criterion quantile criterion 

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Notes

Acknowledgments

This work was supported by the Russian Foundation for Basic Research, project no. 17-07-00203A.

References

  1. 1.
    Birge, J.R. and Louveaux, F., Introduction to Stochastic Programming, New York: Springer, 2011.CrossRefzbMATHGoogle Scholar
  2. 2.
    Kall, P. and Mayer, J., Stochastic Linear Programming: Models, Theory and Computation, New York: Springer, 2011.CrossRefzbMATHGoogle Scholar
  3. 3.
    Pr´ekopa, A., Stochastic Programming, Boston: Kluwer, 1995.CrossRefGoogle Scholar
  4. 4.
    Shapiro, A., Dentcheva, D., and Ruszczy´nski, A., Lectures on Stochastic Programming. Modeling and Theory, Philadelphia: SIAM, 2009.CrossRefGoogle Scholar
  5. 5.
    Wets, R.J.-B., Stochastic Programs with Fixed Recourse: The Equivalent Deterministic Program, SIAM Rev., 1974, vol. 16, no. 3, pp. 309–339.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frauendorfer, K., Stochastic Two-Stage Programming, Berlin—Heidelberg: Springer, 1992.CrossRefzbMATHGoogle Scholar
  7. 7.
    Kulkarni, A.A. and Shanbhag, U.V., Recourse-Based Stochastic Nonlinear Programming: Properties and Benders-SQP Algorithms, Comput. Optim. Appl., 2012, vol. 51, no. 1, pp. 77–123.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kibzun, A.I. and Kan, Y.S., Stochastic Programming Problems with Probability and Quantile Functions, Chichester—New York—Brisbane—Toronto—Singapore: Wiley, 1996.zbMATHGoogle Scholar
  9. 9.
    Kibzun, A.I. and Kan, Y.S., Zadachi stokhasticheskogo programmirovaniya s veroyatnostnymi kriteriyami (Stochastic Programming Problems with Probabilistic Criteria), Moscow: Fizmatlit, 2009.Google Scholar
  10. 10.
    Kibzun, A.I. and Naumov, A.V., A Two-Stage Quantile Linear Programming Problem, Autom. Remote Control, 1995, vol. 56, no. 1, part 1, pp. 68–76.zbMATHGoogle Scholar
  11. 11.
    Schultz, R. and Tiedemann, S., Conditional Value-at-Risk in Stochastic Programs with Mixed-Integer Recourse, Math. Program., Ser. B, 2006, vol. 105, pp. 365–386.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Norkin, V.I., Kibzun, A.I., and Naumov, A.V., Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems, Cybernet. Syst. Anal., 2014, vol. 50, no. 5, pp. 679–692.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sen, S., Relaxation for Probabilistically Constrained Programs with Discrete Random Variables, Oper. Res. Lett., 1992, vol. 11, pp. 81–86.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ruszczy´nski, A., Probabilistic Programming with Discrete Distributions and Precedence Constrained Knapsack Polyhedra, Math. Program., 2002, vol. 93, pp. 195–215.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Luedtke, J., Ahmed, S., and Nemhauser, G., An Interger Programming Approach for Linear Programs with Probabilistic Constraints, Math. Program., 2010, vol. 122, pp. 247–272.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Saxena, A., Goyal, V., and Lejeune, M.A., MIP Reformulations of the Probabilistic Set Covering Problem, Math. Program., 2010, vol. 121, pp. 1–31.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bogdanov, A.B. and Naumov, A.V., Solution to a Two-step Logistics Problem in a Quantile Statement, Autom. Remote Control, 2006, vol. 67, no. 12, pp. 1893–1899.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kibzun, A.I. and Tarasov, A.N., Stochastic Model of the Electric Power Purchase System on a Railway Segment, Autom. Remote Control, 2018, vol. 79, no. 3, pp. 425–438.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Artstein, Z. and Wets, R.J.-B., Consistency of Minimizers and the SLLN for Stochastic Programs, J. Convex Anal., 1996, vol. 2, pp. 1–17.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Pagnoncelli, B.K., Ahmed, S., and Shapiro, A., Sample Average ApproximationMethod for Chance Constrained Programming: Theory and Applications, J. Optim. Theory Appl., 2009, vol. 142, pp. 399–416.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Campi, M.C. and Garatti, S., A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality, J. Optim. Theory Appl., 2011, vol. 148, pp. 257–280.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Higle, J.L. and Sen, S., Statistical Approximations for Stochastic Linear Programming Problems, Ann. Oper. Res., 1999, vol. 85, pp. 173–192.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ivanov, S.V. and Kibzun, A.I., On the Convergence of Sample Approximations for Stochastic Programming Problems with Probabilistic Criteria, Autom. Remote Control, 2018, vol. 79, no. 2, pp. 216–228.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ioffe, A.D. and Tikhomirov, V.M., Theory of Extremal Problems, Amsterdam: North-Holland, 1979.Google Scholar
  25. 25.
    Rockafellar, R.T. and Wets, R.J.-B., Variational Analysis, Berlin: Springer, 2009.zbMATHGoogle Scholar
  26. 26.
    Shiryaev, A.N., Probability-1, New York: Springer, 2016.CrossRefzbMATHGoogle Scholar
  27. 27.
    Zhenevskaya, I.D. and Naumov, A.V., The Decomposition Method for Two-Stage Stochastic Linear Programming Problems with Quantile Criterion, Autom. Remote Control, 2018, vol. 79, no. 2, pp. 229–240.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kibzun, A.I., Comparison of Two Algorithms for Solving a Two-Stage Bilinear Stochastic Programming Problem with Quantile Criterion, Appl. Stochast. Models Business Industry, 2015, vol. 31, no. 6, pp. 862–874.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ivanov, S.V. and Kibzun, A.I., Sample Average Approximation in a Two-Stage Stochastic Linear Program with Quantile Criterion, Proc. Steklov Inst. Math., 2018, vol. 303, no. 1, pp. 107–115.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Moscow Aviation Institute (National Research University)MoscowRussia

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