Annals of Operations Research

, Volume 31, Issue 1, pp 347–369 | Cite as

A numerical method for solving stochastic programming problems with moment constraints on a distribution function

  • Alexei A. Gaivoronski


The stochastic programming problem is considered in the case of a distribution function with partially known random parameters. A minimax approach is taken, and a numerical method is proposed for problems when information on the distribution function can be expressed in the form of finitely many moment constraints. Convergence is proved and results of numerical experiments are reported.

Key words

Stochastic programming incomplete information on distribution function moment constraints stochastic quasigradient methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Birge and R.J.-B. Wets, Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse, Math. Progr. Study 27 (1986) 54–102.Google Scholar
  2. [2]
    J. Birge and R.J.-B. Wets, Computing bounds for stochastic optimization problems by means of a generalized moment problem, Math. Oper. Res. 12 (1987) 149–162.Google Scholar
  3. [3]
    A. Ben-Tal, Stochastic programs with incomplete information. Oper. Res. 24 (1976) 336–347.Google Scholar
  4. [4]
    T. Cipra, Moment problem with given covariance structure stochastic programming. Ekonom.-Mat. Obzor 21 (1985) 66–77.Google Scholar
  5. [5]
    G. Dantzig and A. Madansky, On the solution of two-stage linear programs under uncertainty,Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability, vol. 2 (Univ. California Press. Berkeley, 1961) pp. 165–176.Google Scholar
  6. [6]
    M. Dempster (ed.),Stochastic Programming (Academic Press, London, 1980).Google Scholar
  7. [7]
    J.H. Dulá, An upper bound on the expectation of simplicial functions of multivariate random variables, Math. Progr., to appear.Google Scholar
  8. [8]
    J. Dupačová, Minimax approach to stochastic linear programming and the moment problem. Recent results, ZAMM 58 (1978) T466-T467.Google Scholar
  9. [9]
    J. Dupačová and R.J.-B. Wets, Asymptotic behavior of statistical estimators and optimal solutions for stochastic optimization problems, Working Paper WP-86-112, IIASA (1986).Google Scholar
  10. [10]
    Yu. Ermoliev,Methods of Stochastic Programming (Nauka, Moscow, 1976) (in Russian).Google Scholar
  11. [11]
    Yu. Ermoliev, Stochastic quasigradient methods and their applications to systems optimization, Stochastics 9 (1983) 1–36.Google Scholar
  12. [12]
    Yu. Ermoliev, A. Gaivoronski and C. Nedeva, Stochastic optimization problems with incomplete information on distribution functions, SIAM J. Control Optim. 23 (1985) 697–716.Google Scholar
  13. [13]
    Yu. Ermoliev and V. Norkin, Normalized convergence in stochastic optimization, this volume.Google Scholar
  14. [14]
    Yu. Ermoliev and R.J.-B. Wets (eds.),Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988).Google Scholar
  15. [15]
    K. Frauendorfer, Solving SLP recourse problems with arbitrary multivariate distributions—the dependent case, Math. Oper. Res. 13 (1988) 377–394.Google Scholar
  16. [16]
    K. Frauendorfer, SLP problems: objective and right hand side being stochastically dependent — Part II, Manuscript, University of Zurich (1988).Google Scholar
  17. [17]
    A.A. Gaivoronski, Implementation of stochastic quasigradient methods, in:Numerical Techniques for Stochastic Optimization, eds. Yu. Ermoliev and R.J.-B. Wets (Springer, Berlin, 1988).Google Scholar
  18. [18]
    A.A. Gaivoronski, Interactive program SQG-PC for solving stochastic programming problems on IBM PC/XT/AT compatibles. User guide, Working Paper WP-88-11, IIASA, Laxenburg (1988).Google Scholar
  19. [19]
    A.A. Gaivoronski, Stochastic optimization techniques for finding optimal submeasures, in:Stochastic Optimization eds. V.I. Arkin, A. Shiraev and R. Wets, Lecture Notes in Control and Information Sciences (Springer, Berlin, 1986) pp. 351–363.Google Scholar
  20. [20]
    H. Gassman and W. Ziemba, A tight upper bound for the expectation of a convex function of a multi-variate random variable. Math. Progr. Study 27 (1986) 39–53.Google Scholar
  21. [21]
    A.N. Golodnikov, Solution method for some stochastic problems with partially defined distribution function, Proc. Sov. Acad. Sci., Technical Cybernetics 2 (1980) 19–25 (in Russian).Google Scholar
  22. [22]
    P. Kall,Stochastic Linear Programming (Springer, Berlin, 1976).Google Scholar
  23. [23]
    P. Kall, Stochastic programs with recourse: an upper bound and the related moment problem. Z. Oper. Res. 31 (1987) A119-A141.Google Scholar
  24. [24]
    P. Kall, On approximations and stability in stochastic programming, in:Parametric Optimization and Related Topics eds. J. Guddat, H.Th. Jongen, B. Kummer and F. Nozicka (Akademie-Verlag, Berlin, 1987) pp. 387–407.Google Scholar
  25. [25]
    P. Kall, An upper bound for SLP using first and second moments. this volume.Google Scholar
  26. [26]
    J. Kemperman, On a class of moment problems.Proc. 6th Berkeley Symp. on Mathematical Statistics and Probability, vol. 2 (1972) pp. 101–126.Google Scholar
  27. [27]
    A.J. King, Asymptotic behavior of solutions in stochastic optimization: nonsmooth analysis and the derivation of non-normal limit distributions, Dissertation, University of Washington (1986).Google Scholar
  28. [28]
    M. Krein and A. Nudelman, The Markov moment problem and extremal problems. Transl. Math. Monographs, 50, Amer. Math. Soc., Providence (1977).Google Scholar
  29. [29]
    K. Marti, Descent stochastic quasigradient methods, in:Numerical Techniques for Stochastic Optimization, eds. Yu. Ermoliev and R.J.-B. Wets (Springer, Berlin, 1988).Google Scholar
  30. [30]
    J.L. Nazareth and R.J.-B. Wets, Algorithms for stochastic programs: the case of non-stochastic tenders, Math. Progr. Study 28 (1986) 1–28.Google Scholar
  31. [31]
    E.A. Nurminski,Numerical Methods for Solving Deterministic and Stochastic Minimax Problems (Naukova Dumka, Kiev, 1979).Google Scholar
  32. [32]
    G. Pflug, On the determination of the step-size in stochastic quasigradient methods, Working Paper CP-83-25, IIASA, Laxenburg (1983).Google Scholar
  33. [33]
    B.T. Poliak and Ya.Z. Tsypkin, Robust pseudogradient adaptation algorithms, Automation and Remote Control 41 (1980) 1404–1410.Google Scholar
  34. [34]
    A. Prékopa, Network planning using two-stage programming under uncertainty, in:Recent Results in Stochastic Programming, Lecture Notes in Economics and Mathematical Sciences 179 (Springer. 1980) pp. 215–237.Google Scholar
  35. [35]
    S.M. Robinson and R.J.-B. Wets, Stability in two-stage stochastic programming, SIAM J. Control Optim. 25 (1987).Google Scholar
  36. [36]
    R.T. Rockafellar and R.J.-B. Wets, A Lagrangian finite generation technique for solving linear quadratic problems in stochastic programming, Math. Progr. Study 28 (1986) 63–93.Google Scholar
  37. [37]
    R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970.Google Scholar
  38. [38]
    W. Römisch, On convergence rates of approximations in stochastic programming, Manuskript, Sekt. Math., Humbold-Universität Berlin (1985).Google Scholar
  39. [39]
    A. Ruszczyński and W. Syski, A method of aggregate stochastic subgradients with on-line stepsize rules for convex stochastic programming problems. Math. Progr. Study 28 (1986) 113–131.Google Scholar
  40. [40]
    G. Salinetti, Convergence of stochastic infima: equi-semicontinuity, Tech. Report Nr. 5, Dip. Statist. Probab. Statist. Appl., Università di Roma (1985).Google Scholar
  41. [41]
    R. Van Slyke and R.J.-B. Wets, L-shaped linear programs with applications to optimal control and stochastic programming, SIAM J. Appl. Math. 17 (1969) 638–663.Google Scholar
  42. [42]
    R.J.-B. Wets, Stochastic programming: solution techniques and approximation schemes in:Mathematical Programming: The State of the Art, eds. A. Bachem, M. Grötschel and B. Korte (Springer, 1983) pp. 566–603.Google Scholar
  43. [43]
    J. Ĵáčková, On minimax solutions of stochastic linear programming problems, Časopis pro Pěstováni Matematiky 91 (1966) 423–429.Google Scholar

Copyright information

© J.C. Baltzer A. G. Scientific Publishing Company 1991

Authors and Affiliations

  • Alexei A. Gaivoronski
    • 1
  1. 1.V. Glushkov Institute of CyberneticsKievUSSR

Personalised recommendations