Mathematical Programming

, Volume 81, Issue 3, pp 301–325 | Cite as

A simulation-based approach to two-stage stochastic programming with recourse

  • Alexander Shapiro
  • Tito Homem-de-Mello


In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for two-stage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution. We think that the novelty of the numerical approach developed in this paper is twofold. First, various variance reduction techniques are applied in order to enhance the rate of convergence. Successful application of those techniques is what makes the whole approach numerically feasible. Second, a statistical inference is developed and applied to estimation of the error, validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.


Two-stage stochastic programming with recourse Monte Carlo simulation Likelihood ratios Variance reduction techniques Confidence intervals Hypotheses testing Validation analysis Nonlinear programming 


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  1. [1]
    Y. Ermoliev, Stochastic quasi-gradient methods and their application to systems optimization, Stochastics 4 (1983) 1–37.Google Scholar
  2. [2]
    Y. Ermoliev, R.J.B. Wets (Eds.), Numerical Techniques for Stochastic Optimization. Springer, Berlin, 1988.Google Scholar
  3. [3]
    J.L. Higle, S. Sen, Stochastic decomposition: An algorithm for two-stage linear programs with recourse, Mathematics of Operations Research 16 (1991) 650–669.Google Scholar
  4. [4]
    G. Infanger, Planning under Uncertainty, Solving Large Scale Stochastic Linear Programs, Boyd & Fraser Publishing Company, MAs, USA, 1994Google Scholar
  5. [5]
    E.L. Plambeck, B.R. Fu, S.M. Robinson, R. Suri, Sample-path optimization of convex stochastic performance functions, Mathematical Programming, Series B 75 (1996) 137–176.Google Scholar
  6. [6]
    R.Y. Rubinstein, A. Shapiro, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method, Wiley, New York, 1993.Google Scholar
  7. [7]
    G. Dantzig, Linear programming under uncertainty, Management Science 1 (1955) 197–206.Google Scholar
  8. [8]
    E. Beale, On minimizing a convex function subject to linear inequalities, Journal of the Royal Statistical Society Series B 17 (1955) 173–184.Google Scholar
  9. [9]
    J. Dupačová, Multistage stochastic programs: The state-of-the-art and selected bibliography, Kybernetika 31 (1995) 151–174.Google Scholar
  10. [10]
    P. Kall, S.W. Wallace, Stochastic Programming, Wiley, Chichester, 1994.Google Scholar
  11. [11]
    R. Wets, Stochastic programming: solution techniques and approximation schemes, Mathematical Programming: The State-of-the-Art 1982, Springer, Berlin, 1983 pp. 566–603.Google Scholar
  12. [12]
    R. Wets, Stochastic programs with fixed recourse: the equivalent deterministic program, SIAM Review 16 (1974) 309–339.Google Scholar
  13. [13]
    G. Gürkan, A.Y. Özge, S.M. Robinson, Sample-path optimization in simulation, Proceedings of the 1994 Winter Simulation Conference, 247–254.Google Scholar
  14. [14]
    R.Y. Rubinstein, A. Shapiro, Optimization of static simulation models by the score function method, Mathematics and Computers in Simulation 32 (1990) 373–392.Google Scholar
  15. [15]
    R.T. Rockafellar, R.J.-B. Wets, On the interchange of subdifferentiation and conditional expectation for convex functionals, Stochastics 7 (1982) pp. 173–182.Google Scholar
  16. [16]
    R.Y. Rubinstein, Sensitivity analysis of discrete event systems by the push-out method, Annals of Operations Research 39 (1992) 229–250.Google Scholar
  17. [17]
    R.Y. Rubinstein, A. Shapiro, On optimal choice of reference parameters in the likelihood method, Proceedings of the 1992 Winter Simulation Conference, 1992, pp. 515–520.Google Scholar
  18. [18]
    W. Römisch, R. Schultz, Stability of solutions for stochastic programs with complete recourse. Mathematics of Operations Research 18 (1993) 590–609.Google Scholar
  19. [19]
    A. Shapiro, Y. Wardi, ‘Convergence analysis of stochastic algorithms’, Mathematics of Operations Research 21 (1996) 615–628.Google Scholar
  20. [20]
    R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982.Google Scholar
  21. [21]
    J.L. Higle, S. Sen, Statistical verification of optimality conditions for stochastic programs with recourse, Annals of Operations Reserach 30 (1991) 215–240.Google Scholar
  22. [22]
    A.M. Mathai, S.B. Provost, Quadratic Forms in Random Variables: Theory and Applications, Dekker, New York, 1992.Google Scholar
  23. [23]
    T. Robertson, F.T. Wright, R.L. Dykstra, Order restricted Statistical Inference, Wiley, New York, 1988.Google Scholar
  24. [24]
    A. Shapiro, Towards a unified theory of inequality constrained testing in multivariate analysis, International Statistical Review 56 (1988) 49–62.Google Scholar
  25. [25]
    A. Shapiro, Asymptotic analysis of stochastic programs, Annals of Operations Research 30 (1991) 169–186.Google Scholar
  26. [26]
    A.J. King, R.T. Rockafellar, Asymptotic theory for solutions in statistical estimation and stochastic programming, Mathematics of Operations Research 18 (1993) 148–162.Google Scholar
  27. [27]
    A. Shapiro, Asymptotic behavior of optimal solutions in stochastic programming, Mathematics of Operations Research 18 (1993) 829–845.Google Scholar
  28. [28]
    A.V. Fiacco, G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968.Google Scholar
  29. [29]
    M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley, New York, 1993.Google Scholar
  30. [30]
    P.L'Ecuyer, G. Yin, Budget-dependent convergence rate of stochastic approximation, Preprint.Google Scholar
  31. [31]
    A. Shapiro, Simulation based optimization — convergence analysis and statistical inference, Stochastic Models 12 (1996) 425–454.Google Scholar
  32. [32]
    J.L. Higle, S. Sen, Duality and statistical tests of optimality for two stage stochastic programs, Mathematical Programming, Series B 75 (1996) 257–275.Google Scholar
  33. [33]
    A. Shapiro, Y. Wardi, Nondifferentiability of the steady-state function in Discrete Event Dynamic Systems, IEEE Transactions on Automic Control 39 (1994) 1707–1711.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc 1998

Authors and Affiliations

  • Alexander Shapiro
    • 1
  • Tito Homem-de-Mello
    • 1
  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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