Skip to main content
SpringerLink
Log in

Stochastic Programming: Solution Techniques and Approximation Schemes

  • Chapter
Mathematical Programming The State of the Art

Abstract

Solutions techniques for stochastic programs are reviewed. Particular emphasis is placed on those methods that allow us to proceed by approximation. We consider both stochastic programs with recourse and stochastic programs with chance-constraints.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. M. Beale, On minimizing a convex function subject to linear inequalities, J. Royal Stat. Soc. 17B (1955), 173–184.

    MathSciNet  MATH  Google Scholar 

  2. G. B. Dantzig, Linear programming under uncertainty, Management Sci. 1 (1955), 197–206.

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Tintner, Stochastic linear programming with applications to agricultural economics, in Proc. Second Symposium in Linear Programming, ed. H. A. Antosiewicz, Washington, 1955, 197–207.

    Google Scholar 

  4. A. Charnes and W. W. Cooper, Chance-constrained programming, Management Sci. 5 (1959), 73–79.

    Article  MathSciNet  Google Scholar 

  5. R. Grinold, A class of constrained linear control problems with stochastic coefficients, in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 97–108.

    Google Scholar 

  6. W. Klein Haneveld, A dual of a dynamic inventory control model: the deterministic and stochastic case, in Recent Results in Stochastic Programming, ed. P. Kali and A. Prékopa, Springer Verlag Lecture Notes in Economics and Mathematical Systems, 179 (1980), 67–98.

    Google Scholar 

  7. J. Dupacová, Water resource systems using stochastic programming with recourse, in Recent Results in Stochastic Programming, ed. P. Kali and A. Prékopa, Springer- Verlag Lecture Notes in Economics and Mathematical Systems, 179 (1980), 121–134.

    Google Scholar 

  8. K. Back, Optimality and equilibrium in infinite horizon economies under uncertainty, Ph. D. thesis, University of Kentucky, Lexington, 1982.

    Google Scholar 

  9. R. Everitt and W. T. Ziemba, Two-period stochastic programmes with simple recourse, Operations Research 27 (1979), 485–502.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Dempster, Stochastic Programming, Academic Press, London, 1980.

    MATH  Google Scholar 

  11. D. Walkup and R. J.-B. Wets, Stochastic programs with recourse, SIAM J. AppL Math. 15 (1967), 316–339.

    Article  MathSciNet  Google Scholar 

  12. P. Olsen, Multistage stochastic programming with recourse: the equivalent deterministic program, SIAM J. Control Optim. 14 (1976), 495–517.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Wets, Stochastic programs with fixed recourse: the equivalent deterministic program, SIAM Review 16 (1974), 309–339.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Kali and W. Oettli, Measurability theorems for stochastic extremals, SIAM J. Control Optim. 13 (1975), 994–998.

    Article  Google Scholar 

  15. R. Wets, Induced constraints for stochastic optimization problems, in Techniques of Optimization, ed. A. Balakrishnan, Academic Press, London, 1972, 433–443.

    Google Scholar 

  16. A. Prékopa, Logarithmic concave measures with applications to stochastic programming, Acta Sci. Math. 32 (1971), 301–316.

    MATH  Google Scholar 

  17. A. Prékopa, Logarithmic concave measures and related topics, in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 63–82.

    Google Scholar 

  18. J. G. Kallberg and W. T. Ziemba, Generalized concave functions in stochastic programming and portfolio theory, in Generalized Concavity in Optimization and Economics, ed. S. Schaible and W. Ziemba, Academic Press, New York, 1981, 719– 767.

    Google Scholar 

  19. Chr. Borell, Convex set functions in d-spaces, Period. Math. Hungar. 6 (1975), 111–136.

    Article  MathSciNet  Google Scholar 

  20. C. Van de Panne and W. Popp, Minimum-cost cattle feed under probabilistic protein constraints, Management Sci. 9 (1963), 405–430.

    Article  Google Scholar 

  21. A. Prékopa, Programming under probabilistic constraints with a random technology matrix, Math. Operationsforsch. Statist. 5 (1974), 109–116.

    Article  MathSciNet  Google Scholar 

  22. S. M. Sinha, Stochastic Programming, Doc. Thesis, University of California-Berkeley, 1963; see also Proceed. Symposium on Probability and Statistics, Ben Hindu University, India.

    Google Scholar 

  23. R. T. Rockafellar, Integrals which are convex functional II, Pacific J. Mathematics 39 (1971), 439–469.

    MathSciNet  MATH  Google Scholar 

  24. R. T. Rockafellar and R. Wets, On the interchange of subdifferentiation and conditional expectation for convex functionals, Stochastics 7 (1982), 173–182.

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Walkup and R. Wets, Stochastic programs with recourse: special forms, in Proceedings of the Princeton Symposium on Mathematical Programming, ed. H. Kuhn, Princeton University Press, Princeton, 1970, 139–162.

    Google Scholar 

  26. A. Prékopa, Network planning using two-stage programming under uncertainty, in Recent Results in Stochastic Programming, eds. P. Kail and A. Prékopa, Springer- Verlag Lecture Notes in Economics and Mathematical Systems, Vol. 179, 1980, 215–237.

    Google Scholar 

  27. A. Prékopa, S. Ganczer, I. Deák and K. Patyi, The STABIL stochastic programming model and its experimental application to the electrical energy sector of the Hungarian economy, in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 369–385.

    Google Scholar 

  28. R. Wets, Programming under uncertainty: the equivalent convex program, SIAM J. Appl. Math. 14 (1966), 89–105.

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Hansotia, Stochastic linear programs with simple recourse: the equivalent deterministic convex program for the normal, exponential and Erlang cases, Naval Res. Logist. Quat. 27 (1980), 257–272.

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Nazareth and R. Wets, Algorithms for stochastic programs: the case of nonsto- chastic tenders, II AS A Working Paper, Laxenburg, Austria, 1982.

    Google Scholar 

  31. F. Louveaux, A solution method for multistage stochastic programs with recourse with application to an energy investment problem, Operations Res. 28 (1980), 889– 902.

    Google Scholar 

  32. J. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Tech. Report 82-6, Dept. Industrial and Operations Engineering, University of Michigan, 1982.

    Google Scholar 

  33. B. Strazicky, Some results concerning an algorithm for the discrete recourse problem in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 263–274.

    Google Scholar 

  34. P. Kali, Z. Angew. Math. Phys. 30 (1979), 261–271; see also Large-Scale Linear Programming, eds. G. Dantzig, M. Dempster and M. Kallio, IIASA Collaborative Proceedings Series, Laxenburg, Austria, 1981, 287–298.

    Google Scholar 

  35. P. Kali and D. Stoyan, Solving stochastic programming problems with recourse including error bounds, Math. Operationsforsch. Statist. Ser. Optimization (1982).

    Google Scholar 

  36. G. Dantzig and A. Madansky, On the solution of two-stage linear programs under uncertainty, Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, 1961, 165–176.

    Google Scholar 

  37. R. Van Slyke and R. Wets, L-shaped linear programs with applications to optimal control and stochastic programming, SIAM J. Appl. Math. 17 (1969), 638–663.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Birge, Solution methods for stochastic dynamic linear programs, Tech. Report SOL 80-29, Systems Optimization Laboratory, Stanford University, 1980.

    Google Scholar 

  39. S. Gartska and D. Ruthenberg, Computation in discrete stochastic programs with recourse, Operations Res. 21 (1973), 112–122.

    Article  MathSciNet  Google Scholar 

  40. R. Wets, Characterization theorems for stochastic programs, Math. Programming 2 (1972), 166–175.

    Article  MathSciNet  MATH  Google Scholar 

  41. Y. Ermoliev, Stochastic quasigradient methods and their application in systems optimization, IIASA Working Paper, WP-81-2, Laxenburg, Austria, 1981; to appear in Stochastics.

    Google Scholar 

  42. Y. Ermoliev, Stochastic Programming Methods (in Russian), in Nauka, Moscow 1976.

    Google Scholar 

  43. Y. Ermoliev and E. Nurminski, Stochastic quasigradient algorithms for minimax problems in stochastic programming, in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 275–285.

    Google Scholar 

  44. B. Poljak, Nonlinear programming methods in the presence of noise, Math. Programming 14 (1978), 87–97.

    Article  MathSciNet  MATH  Google Scholar 

  45. K. Marti, Approximationen Stochastischer Optimierungsprobleme, Verlag Anton Hain, Kônigstein, 1979.

    MATH  Google Scholar 

  46. Y. Ermoliev, G. Leonardi and J. Vira, The stochastic quasigradient method applied to a facility location problem, IIASA Working Paper, WP-81-14, Laxenburg, Austria, 1981.

    Google Scholar 

  47. J. Dodu, M. Goursat, A. Hertz, J.-P. Quadrat and M. Viot, Méthodes de gradient stochastique pour l’optimisation des investissements dans un résau électrique, E. D. F. Bulletin-Série C, (1981), No. 2, 133–164.

    MathSciNet  Google Scholar 

  48. P. Kall, Approximations to stochastic programs with complete fixed recourse, Numer. Math. 22 (1974), 333–339.

    Article  MathSciNet  MATH  Google Scholar 

  49. P. Olsen, Discretizations of multistage stochastic programming problems, Mathematical Programming Study 6 (1976), 111–124; also Ph. D. Thesis, Cornell Univ., 1974.

    Google Scholar 

  50. G. Salinetti, Approximations for chance-constrained programming problems, Tech. Report No. 13, Istituto della Probabilità, Univ. Roma, 1981; to appear in Stochastics (1983).

    Google Scholar 

  51. R. Wets, Solving stochastic programs with simple recourse II, in Proceed. 1975 Conference on Information Sciences and Systems, The Johns Hopkins University, Baltimore, 1975.

    Google Scholar 

  52. A. Dexter, J. Yu and W. Ziemba, Portfolio selection in a lognormal market when the investor has a power utility function: computational results, in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 507–523.

    Google Scholar 

  53. R. Grinold, A new approach to multistage stochastic linear programs, Mathematical Programming Study 6 (1976), 19–29.

    MathSciNet  Google Scholar 

  54. E. Beale, J. Forrest and C. Taylor, Multi-time-period stochastic programming, in Stochastic Programming, ed. M. Dempster, Academic Press, London, 1980, 387–402.

    Google Scholar 

  55. M. Queyranne and E. Kao, Aggregation in a two-stage stochastic program for manpower planning in the service sector, Tech. Report, University of Houston, 1981.

    Google Scholar 

  56. E. G. Read, Approaches to stochastic reservoir modelling, Working Paper No. 153, College Business Admin., University of Tennessee, 1982.

    Google Scholar 

  57. K. Marti, Approximationen von Entscheidungsproblemen mit linearer Ergebnisfunktion und positiv homogener, subadditiver Verlustfunktion, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 31 (1975), 203–233.

    Article  MathSciNet  MATH  Google Scholar 

  58. W. Römisch, On discrete approximation in stochastic programming, Proceed. 13. Jahrestagung Mathematische Optimierung, Vitte 1981, Humboldt-Univ., Berlin, Sektion Mathematik, Seminarbericht 39, 1981.

    Google Scholar 

  59. B. Van Cutsem, Problems of convergence in stochastic linear programming, in Techniques of Optimization, ed. A. Balakrishnan, Academic Press, New York, 1972, 445–454.

    Google Scholar 

  60. H. Attouch and R. Wets, Convergence and approximation in nonlinear optimization, in Nonlinear Programming 4, eds. O. Mangasarian, R. Meyer and S. Robinson, Academic Press, New York, 1981, 367–394.

    Google Scholar 

  61. A. Madansky, Inequalities for stochastic linear programming problems, Management Sci. 6 (1960), 197–204.

    Article  MathSciNet  MATH  Google Scholar 

  62. R. Hartley, Inequalities in completely convex stochastic programming, J. Math. Anal. Applic. 75 (1980), 373–384.

    Article  MathSciNet  MATH  Google Scholar 

  63. C. Huang, W. Ziemba and A. Ben-Tal, Bounds on the expectation of a convex function of a random variable: with application to stochastic programming, Operations Res. 25 (1977), 315–325.

    Article  MathSciNet  MATH  Google Scholar 

  64. C. Huang, I. Vertinsky and W. Ziemba, Sharp bounds on the value of perfect information, Operations Res. 25 (1977), 128–139.

    Article  MathSciNet  MATH  Google Scholar 

  65. M. Iosifescu and R. Theodorescu, Linear programming under uncertainty, in Colloquium on Applications of Mathematics to Economics, Budapest, 1963, ed. A. Prékopa, Publishing House Hungarian Academy of Sciences, Budapest, 1965, 133 - 140.

    Google Scholar 

  66. J. Zacková, On minimax solutions of stochastic linear programming problems, Casopis Pest. Mat. 91 (1966), 423–429.

    MathSciNet  MATH  Google Scholar 

  67. J. Dupacova, On minimax decision rule in stochastic linear programming, Math. Methods Oper. Res. 1 (1980), 47–60.

    MathSciNet  Google Scholar 

  68. J. Dupacova, Minimax approach to stochastic linear programming and the moment problem, (in Czech), Ekonomickomatematicky Ohzor 13 (1977), 279–307. Summary of results in ZAMM 58 (1978), T466–T467.

    MathSciNet  Google Scholar 

  69. A. Williams, Approximation formulas for stochastic linear programming, SIAM J. Appl. Math. 14 (1966), 668–677.

    Article  MathSciNet  MATH  Google Scholar 

  70. J. Birge, The value of the stochastic solution in stochastic linear programs with fixed recourse, Math. Programming, 1983.

    Google Scholar 

  71. J. Birge and R. Wets, On initial solutions and approximation schemes for stochastic programs with recourse, IIASA Worling Paper, Laxenburg, Austria, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, Kentucky, 40506, USA

    R. Wets

  2. I.IA.S.A, 2361, Laxenburg, Austria

    R. Wets

Authors
  1. R. Wets
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Ökonometrie und Operations Research, Universität Bonn, Nassestraße 2, D-5300, Bonn 1, Germany

    Achim Bachem  & Bernhard Korte  & 

  2. Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, Memminger Straße 6, D-8900, Augsburg, Germany

    Martin Grötschel

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Wets, R. (1983). Stochastic Programming: Solution Techniques and Approximation Schemes. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-68874-4_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-68876-8

  • Online ISBN: 978-3-642-68874-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation