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Bound‐based approximations in multistage stochasticprogramming: Nonanticipativity aggregation

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Abstract

This paper considers bounding techniques within multistage stochastic programmingand their computational issues. First, a brief overview of bound‐based approximations isprovided within the context of two‐stage stochastic programs. Then, a new methodology forapproximating multistage stochastic programs is developed using a process called non‐anticipativityaggregation. The method advocates relaxing nonanticipativity requirementsof the decisions up to a certain second‐order aggregation to compute lower bounds. Then,this relaxation is refined progressively towards solving the stochastic program. Preliminarycomputational results are presented.

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References

  1. J.R. Birge, An L-shaped method computer code for multistage linear programs, in: Numerical Techniques for Stochastic Optimization, eds. Y. Ermoliev and R.J-B Wets, Springer, 1988.

  2. J.R. Birge, C.J. Donohue, D.F. Holmes and O.G. Svintsitsli, A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs, Technical Report 94–1, Department of Industrial and Operations Engineering, University of Michigan, 1994.

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

    Google Scholar 

  4. 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.

    Google Scholar 

  5. J.R. Birge and R.J-B Wets, Sublinear upper bounds for stochastic programs with recourse, Mathematical Programming 43(1989)131–149.

    Google Scholar 

  6. D.R. Cariño, D.H. Myers, and W.T. Ziemba, Concepts, technical issues, and uses of the Russell-Yasuda Kasai financial planning model, Operations Research 46(1998)450–462.

    Google Scholar 

  7. G.B. Dantzig and P.W. Glynn, Parallel processors for planning under uncertainty, Annals of Operations Research 22(1990)1–21.

    Google Scholar 

  8. M.A.H. Dempster, The expected value of perfect information in the optimal evolution of stochastic systems, in: Stochastic Differential Systems, eds. M. Arato, D. Vermes, A.V. Balakrishnan, Lecture Notes in Control and Information Sciences, 1981, pp. 25–40.

  9. J.H. Dula, An upper bound on the expectation of simplicial functions of multivariate random variables, Mathematical Programming 55(1992)69–80.

    Google Scholar 

  10. J. Dupačová (as Žáčková), On minimax solutions of stochastic linear programming problems, Časopis Pro Pětováni Matematiky 91(1966)423–430.

    Google Scholar 

  11. J. Dupačová, Minimax stochastic programs with nonconvex nonseparable penalty functions, in: Progress in Operations Research, ed. A. Prékopa, North-Holland, Amsterdam, 1976, pp. 303–316.

    Google Scholar 

  12. J. Dupačová, Multistage stochastic programs: The state-of-the-art and selected bibliography, Kybernetika 2(1995)151–174.

    Google Scholar 

  13. J. Dupačová, Scenario-based stochastic programs: Resistance with respect to sample, Annals of Operations Research 64(1996)21–38.

    Google Scholar 

  14. N.C.P. Edirisinghe, New second-order bounds on the expectation of saddle functions with applications to stochastic linear programming, Operations Research 44(1996)909–922.

    Google Scholar 

  15. N.C.P. Edirisinghe, Partitioning strategies for scenario approximation in SLP, Working paper, University of Tennessee, 1998.

  16. N.C.P. Edirisinghe and W.T. Ziemba, Tight bounds for stochastic convex programs, Operations Research 40(1992)660–677.

    Google Scholar 

  17. N.C.P. Edirisinghe and W.T. Ziemba, Bounds for two-stage stochastic programs with fixed recourse, Mathematics of Operations Research 19(1994)292–313.

    Google Scholar 

  18. N.C.P. Edirisinghe and W.T. Ziemba, Bounding the expectation of a saddle function with application to stochastic programming, Mathematics of Operations Research 19(1994)314–340.

    Google Scholar 

  19. N.C.P. Edirisinghe and W.T. Ziemba, Implementing bounds-based approximations in convex-concave two-stage stochastic programming, Mathematical Programming 19(1996)314–340.

    Google Scholar 

  20. N.C.P. Edirisinghe and G-M. You, Second-order scenario approximation and refinement in optimization under uncertainty, Annals of Operations Research 64(1996)143–178.

    Google Scholar 

  21. N.C.P. Edirisinghe and G-M. You, Scenario approximation in multistage stochastic programming: The case of nonstochastic tenders, Working paper, University of Tennessee, 1998.

  22. Y. Ermoliev, Stochastic quasigradient methods and their application to systems optimization, Stochastics 9(1983)1–36.

    Google Scholar 

  23. Y. Ermoliev and A. Gaivoronsky, Stochastic quasigradient methods and their implementation, in: Numerical Techniques for Stochastic Optimization, eds., Y. Ermoliev and R.J-B Wets, Springer, 1988.

  24. Y. Ermoliev and R.J-B Wets, Numerical Techniques for Stochastic Optimization, Springer, 1988.

  25. K. Frauendorfer, SLP problems: Objective and right-hand side being stochastically dependent — Part II, Manuscript, IOR, University of Zurich, 1988.

  26. K. Frauendorfer, Stochastic Two-Stage Programming, Springer, Berlin, 1992.

    Google Scholar 

  27. K. Frauendorfer, Barycentric scenario trees in convex multistage stochastic programming, Working Paper, University of St. Gallen, 1993.

  28. K. Frauendorfer, Multistage stochastic programming: Error analysis for the convex case, Zeitschrift für Operations Research 39(1994)93–122.

    Google Scholar 

  29. K. Frauendorfer and P. Kall, A solution method for SLP problems with arbitrary multivariate distributions — The independent case, Problems of Control and Information Theory 17(1988)177–205.

    Google Scholar 

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

    Google Scholar 

  31. H.I. Gassmann and S.W. Wallace, Solving linear programs with multiple right-hand sides: Pricing and ordering schemes, Annals of Operations Research 64(1996)237–259.

    Google Scholar 

  32. H.I. Gassmann and W.T. Ziemba, A tight upper bound for the expectation of a convex function of a multivariate random variable, Mathematical Programming Study 27(1986)39–53.

    Google Scholar 

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

    Google Scholar 

  34. C.C. Huang, W.T. 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.

    Google Scholar 

  35. G. Infanger, Monte Carlo (Importance) sampling within a Benders decomposition algorithm for stochastic linear programs extended version: including results of large-scale programs, Technical Report SOL 91–6, Stanford University, 1991.

  36. P. Kall, An upper bound for SLP using first and total second moments, Annals of Operations Research 30(1991)267–276.

    Google Scholar 

  37. P. Kall, Stochastic programming with recourse: Upper bounds and moment problems, A review, in: Advances in Mathematical Optimization, eds. J. Guddat et al., Akademie Verlag, Berlin, 1988.

    Google Scholar 

  38. P. Kall and D. Stoyan, Solving stochastic programming problems with recourse including error bounds, Mathematische Operationsforschung und Statistik, Series Optimization 13(1982)431–447.

    Google Scholar 

  39. A. Madansky, Bounds on the expectation of a convex function of a multivariate random variable, Annals of Mathematical Statistics 30(1959)743–746.

    Google Scholar 

  40. D.P. Morton and R.K. Wood, Restricted-recourse bounds for stochastic linear programming, Operations Research (1998), to appear.

  41. J.M. Mulvey and H. Vladimirou, Applying the progressive hedging algorithm to stochastic generalized networks, Annals of Operations Research 31(1991)399–424.

    Google Scholar 

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

    Google Scholar 

  43. A. Ruszczyński, A regularized decomposition method for minimizing a sum of polyhedral functions, Mathematical Programming 35(1986)309–333.

    Google Scholar 

  44. A. Ruszczyński, Parallel decomposition of multistage stochastic programming, Mathematical Programming 58(1993)201–228.

    Google Scholar 

  45. R. Van Slyke and R.J-B Wets, L-shaped linear programs with application to optimal control and stochastic optimization, SIAM Journal on Applied Mathematics 17(1969)638–663.

    Google Scholar 

  46. H. Vladimirou, R.J-B Wets and S.A. Zenios (eds.), Models for planning under uncertainty, Annals of Operations Research 59(1995).

  47. S.W. Wallace, A piecewise linear upper bound on the network recourse function, Mathematical Programming 38(1987)133–146.

    Google Scholar 

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Edirisinghe, N. Bound‐based approximations in multistage stochasticprogramming: Nonanticipativity aggregation. Annals of Operations Research 85, 103–127 (1999). https://doi.org/10.1023/A:1018961525394

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