Skip to main content
SpringerLink
Log in

A gaussian upper bound for gaussian multi-stage stochastic linear programs

  • Published:
Mathematical Programming Submit manuscript

Abstract

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E.M.L. Beale, “On minimizing a convex function subject to linear inequalities,”Journal of the Royal Statistical Society, Series B 17 (1955) 173–184.

    MATH  MathSciNet  Google Scholar 

  2. E.M.L. Beale, G.B. Dantzig and R.D. Watson, “A first order approach to a class of multi-time-period stochastic programming problems,”Mathematical Programming Study 27 (1986) 103–117.

    MATH  MathSciNet  Google Scholar 

  3. E.M.L. Beale, J.J.H. Forrest and C.J. Taylor. “Multi-time-period stochastic programming,” in: M. Dempster, ed.,Stochastic Programming (Academic Press, New York, 1980) pp. 387–402.

    Google Scholar 

  4. P. Billingsley,Probability and Measure (Wiley, New York, 1986).

    MATH  Google Scholar 

  5. J.R. Birge, “Decomposition and partitioning methods for multi-stage stochastic linear programs,”Operations Research 33 (1985) 989–1007.

    MATH  MathSciNet  Google Scholar 

  6. J.R. Birge, “Aggregation bounds in stochastic linear programming,”Mathematical Programming 31 (1985) 25–41.

    Article  MATH  MathSciNet  Google Scholar 

  7. J.R. Birge and S.W. Wallace, “A separable precewise linear upper bound for stochastic linear programs,”SIAM Journal on Control and Optimization 26 (3) (1988) 725–739.

    Article  MATH  MathSciNet  Google Scholar 

  8. J.R. Birge and R.J.-B. Wets, “Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse,”Mathematical Programming Study 27 (1986) 54–102.

    MATH  MathSciNet  Google Scholar 

  9. G.R. Bitran and D. Sarkar, “On upper bounds of sequential stochastic production planning problems,”European Journal of Operational Research 34 (1988) 191–207.

    Article  MATH  MathSciNet  Google Scholar 

  10. G.R. Bitran and H.H. Yanasse, “Deterministic approximations to stochastic production problems,”Operations Research 32 (5) (1984) 999–1018.

    MATH  MathSciNet  Google Scholar 

  11. G.B. Dantzig, “Linear programming under uncertainty,”Management Science 1 (1955) 197–206.

    MATH  MathSciNet  Google Scholar 

  12. G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).

    MATH  Google Scholar 

  13. G.B. Dantzig and G. Infanger, “Multi-stage stochastic linear programs for portfolio optimization,” Technical Report SOL 91-11. Department of Operations Research, Stanford University, Stanford, CA. 1991.

    Google Scholar 

  14. R. Durret,Probability: Theory and Examples (Wadsworth, Pacific Grove, CA, 1991).

    Google Scholar 

  15. R. Entriken and G. Infanger, “Decomposition and importance sampling for stochastic linear models,”Energy, The International Journal 15 (7/8) (1990) 645–659.

    Google Scholar 

  16. S.J. Garstka and R.J.-B. Wets, “On decision rules in stochastic programming,”Mathematical Programming 7 (1974) 117–143.

    Article  MATH  MathSciNet  Google Scholar 

  17. H.I. Gassman, “MSLiP: A computer code for the multistage stochastic linear programming problem,”Mathematical Programming 47 (1990) 407–423.

    Article  MathSciNet  Google Scholar 

  18. J.L. Higle and S. Sen, “Stochastic decomposition: An algorithm for two-stage linear programs with recourse,”Mathematics of Operations Research 16 (1991) 650–669.

    MATH  MathSciNet  Google Scholar 

  19. J. Higle, S. Sen and D.S. Yakowitz, “Finite master programs in stochastic decomposition,” Technical Report. Department of Systems and Industrial Engineering. The University of Arizona, Tucson, AZ. 1990.

    Google Scholar 

  20. J.K. Ho and E. Loute, “A set of staircase linear programming test problems,”Mathematical Programming 20 (1981) 245–250.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. Huang, W. Ziemba and A. Ben-Tal, “Bounds on the expectation of a convex function of a random variable: With applications to stochastic programming,”Operations Research 25 (1977) 315–325.

    MATH  MathSciNet  Google Scholar 

  22. G. Infanger, “Monte Carlo (importance) sampling within a Benders’ decomposition algorithm for stochastic linear programs, Extended version: Including results of large-scale problems,”Annals of Operations Research 39 (1991) 69–95.

    Article  MathSciNet  Google Scholar 

  23. G. Infanger,Planning under uncertainty: Solving large-scale stochastic linear programs (Boyd & Fraser, Danvers, MA, 1994).

    MATH  Google Scholar 

  24. I. Karatzas and S.E. Shreve,Brownian Motion and Stochastic Calculus (Springer-Verlag, New York, 1991).

    MATH  Google Scholar 

  25. F. Louveaux and N. Smeers, “Optimal investment for electricity generation A stochastic model and a test problem,” in: Y. Ermoliev and R.J.-B. Wets,Numerical Techniques for Stochastic Optimization (Springer-Verlag, Berlin, 1988).

    Google Scholar 

  26. A. Madansky, “Inequalities for stochastic linear programming problems,”Management Science 6 (2) (1960) 197–204.

    MATH  MathSciNet  Google Scholar 

  27. H.M. Markowitz,Portfolio Selection: Efficient Diversification of Investments (Basil Blackwell, Cambridge, MA, 1991).

    Google Scholar 

  28. J.M. Mulvey and H. Vladimirou, “Stochastic network optimization models for investment planning,”Annals of Operations Research 20 (1989) 187–217.

    Article  MATH  MathSciNet  Google Scholar 

  29. R.T. Rockafellar and R.J.-B. Wets, “Scenarios and policy aggregation in optimization under uncertainty,”Mathematics of Operations Research 16 (1991) 119–147.

    MATH  MathSciNet  Google Scholar 

  30. A. Ruszczynski, “Parallel decomposition of multistage stochastic programming problems,”Mathematical Programming 58 (1993) 201–228.

    Article  MATH  MathSciNet  Google Scholar 

  31. R.J. Vanderbei and B. Yang, “The simplest semidefinite programs are trivial,” Technical Report SOR-93-12, Program in Statistics and Operations Research, Princeton University, Princeton, NJ, 1993.

    Google Scholar 

  32. S.W. Wallace and T. Yan, “Bounding multi-stage stochastic programs from above,”Mathematical Programming 61 (1993) 111–129.

    Article  MathSciNet  Google Scholar 

  33. R.J.-B. Wets, “Stochastic programming,” in: G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd eds.,Handbook in Operations Research and Management Science, Vol. 1 (Elsevier Science Publishers, Amsterdam, 1989) pp. 573–629.

    Google Scholar 

  34. S.E. Wright, “Primal-dual aggregation and disaggregation for stochastic linear programs,”Mathematics of Operations Research 19 (4) (1994) 893–908.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Industrial Engineering and Management. Technion — Israel Institute of Technology, 32000, Haifa, Israel

    Eithan Schweitzer & Mordecai Avriel

Authors
  1. Eithan Schweitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mordecai Avriel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schweitzer, E., Avriel, M. A gaussian upper bound for gaussian multi-stage stochastic linear programs. Mathematical Programming 77, 1–21 (1997). https://doi.org/10.1007/BF02614515

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02614515

Keywords

Search

Navigation