Continuous Optimization pp 111-146

Part of the Applied Optimization book series (APOP, volume 99)

On Complexity of Stochastic Programming Problems

  • Alexander Shapiro
  • Arkadi Nemirovski


The main focus of this paper is in a discussion of complexity of stochastic programming problems. We argue that two-stage (linear) stochastic programming problems with recourse can be solved with a reasonable accuracy by using Monte Carlo sampling techniques, while multistage stochastic programs, in general, are intractable. We also discuss complexity of chance constrained problems and multistage stochastic programs with linear decision rules.

Key words

stochastic programming complete recourse chance constraints Monte Carlo sampling SAA method large deviations bounds convex programming multi-stage stochastic programming 


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  1. [ADEH99]
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Mathematical Finance, 9, 203–228 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. [ADEHK03]
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. Ku, H.: Coherent multiperiod risk measurement, Manuscript, ETH Zürich (2003)Google Scholar
  3. [BL97]
    Barmish, B.R., Lagoa, C.M.: The uniform distribution: a rigorous justification for the use in robustness analysis. Math. Control, Signals, Systems, 10, 203–222 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. [Bea55]
    Beale, E.M.L.: On minimizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society, Series B, 17, 173–184 (1955)MATHMathSciNetGoogle Scholar
  5. [BN98]
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Mathematics of Operations Research, 23 (1998)Google Scholar
  6. [BN01]
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. SIAM, Philadelphia (2001)MATHGoogle Scholar
  7. [BGGN04]
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear Programs. Mathematical Programming, 99, 351–376 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. [BGNV04]
    Ben-Tal, A., Golany, B., Nemirovski, A., Vial J.-Ph.: Retailer-supplier flexible commitments contracts: A robust optimization approach. Submitted to Manufacturing & Service Operations Management (2004)Google Scholar
  9. [CC05]
    Calafiore G., Campi, M.C.: Uncertain convex programs: Randomized solutions and confidence levels. Mathematical Programming, 102, 25–46 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. [CC04]
    Calafiore, G., Campi, M.C.: Decision making in an uncertain environment: the scenariobased optimization approach. Working paper (2004)Google Scholar
  11. [CC59]
    Charnes, A., Cooper, W.W.: Uncertain convex programs: randomized solutions and confidence levels. Management Science, 6, 73–79 (1959)MathSciNetMATHGoogle Scholar
  12. [DLMV88]
    Dagum, P., Luby, L., Mihail, M., Vazirani, U.: Polytopes, Permanents, and Graphs with Large Factors. Proc. 27th IEEE Symp. on Fondations of Comput. Sci. (1988)Google Scholar
  13. [Dan55]
    Dantzig, G.B.: Linear programming under uncertainty. Management Science, 1, 197–206 (1955)MATHMathSciNetCrossRefGoogle Scholar
  14. [DZ98]
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer-Verlag, New York, NY (1998)MATHGoogle Scholar
  15. [Dup79]
    Dupačová, J.: Minimax stochastic programs with nonseparable penalties. In: Optimization techniques (Proc. Ninth IFIP Conf., Warsaw, 1979), Part 1, 22 of Lecture Notes in Control and Information Sci., 157–163. Springer, Berlin (1980)Google Scholar
  16. [Dup87]
    Dupačová, J.: The minimax approach to stochastic programming and an illustrative application. Stochastics, 20, 73–88 (1987)MathSciNetMATHGoogle Scholar
  17. [DS03]
    Dyer, M., Stougie, L.: Computational complexity of stochastic programming problems. SPOR-Report 2003-20, Dept. of Mathematics and Computer Sci., Eindhoven Technical Univ., Eindhoven (2003)Google Scholar
  18. [ER05]
    Eichhorn, A., Römisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optimization, to appear (2005)Google Scholar
  19. [EGN85]
    Ermoliev, Y., Gaivoronski, A., Nedeva, C: Stochastic optimization problems with partially known distribution functions. SIAM Journal on Control and Optimization, 23, 697–716 (1985)MathSciNetCrossRefMATHGoogle Scholar
  20. [FS02]
    Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance and Stochastics, 6, 429–447 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. [KSH01]
    Kleywegt, A.J., Shapiro, A., Homem-De-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM Journal of Optimization, 12, 479–502 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. [Gai91]
    Gaivoronski, A.A.: A numerical method for solving stochastic programming problems with moment constraints on a distribution function. Annals of Operations Research, 31, 347–370 (1991)MATHMathSciNetCrossRefGoogle Scholar
  23. [JV96]
    Jerrum, M., Vazirani, U.: A mildly exponential approximation algorithm for the permanent. Algorithmica, 16, 392–401 (1996)MathSciNetMATHGoogle Scholar
  24. [LSW05]
    Linderoth, J., Shapiro, A., Wright, S.: The empirical behavior of sampling methods for stochastic programming. Annals of Operations Research, to appear (2005)Google Scholar
  25. [LSW00]
    Linial, N., Samorodnitsky, A., Wigderson, A.: A deterministic strongly poilynomial algorithm for matrix scaling and approximate permanents. Combinatorica, 20, 531–544 (2000)MathSciNetCrossRefGoogle Scholar
  26. [MMW99]
    Mak, W.K., Morton, D.P., Wood, R.K.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Operations Research Letters, 24, 47–56 (1999)MathSciNetCrossRefMATHGoogle Scholar
  27. [Mar52]
    Markowitz, H.M.: Portfolio selection. Journal of Finance, 7, 77–91 (1952)Google Scholar
  28. [Lan87]
    H.J. Landau (ed): Moments in mathematics. Proc. Sympos. Appl. Math., 37. Amer. Math. Soc., Providence, RI (1987)Google Scholar
  29. [Nem03]
    Nemirovski, A.: On tractable approximations of randomly perturbed convex constraints — Proceedings of the 42nd IEEE Conference on Decision and Control. Maui, Hawaii USA, December 2003, 2419–2422 (2003)Google Scholar
  30. [NS05]
    Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints. In: Calafiore, G., Dabbene, F., (eds) Probabilistic and Randomized Methods for Design under Uncertainty. Springer, Berlin (2005)Google Scholar
  31. [Pre95]
    Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht, Boston (1995)Google Scholar
  32. [Rie03]
    Riedel, F.: Dynamic coherent risk measures. Working Paper 03004, Department of Economics, Stanford University (2003)Google Scholar
  33. [RUZ02]
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Deviation measures in risk analysis and optimization, Research Report 2002-7, Department of Industrial and Systems Engineering, University of Florida (2002)Google Scholar
  34. [RS04a]
    Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. E-print available at: (2004)Google Scholar
  35. [RS04b]
    Ruszczyński, A., Shapiro, A.: Conditional risk mappings. E-print available at: (2004)Google Scholar
  36. [SAGS05]
    Santoso, T., Ahmed, S., Goetschalckx, M., Shapiro, A.: A stochastic programming approach for supply chain network design under uncertainty. European Journal of Operational Research, 167, 96–115 (2005)MathSciNetCrossRefMATHGoogle Scholar
  37. [SH00]
    Shapiro, A., Homem-de-Mello, T.: On rate of convergence of Monte Carlo approximations of stochastic programs. SIAM Journal on Optimization, 11, 70–86 (2000)MathSciNetCrossRefMATHGoogle Scholar
  38. [SK00]
    Shapiro, A., Kleywegt, A.: Minimax analysis of stochastic programs. Optimization Methods and Software, 17, 523–542 (2002)MathSciNetCrossRefMATHGoogle Scholar
  39. [SHK02]
    Shapiro, A., Homem de Mello, T., Kim, J.C.: Conditioning of stochastic programs. Mathematical Programming, 94, 1–19 (2002)MathSciNetCrossRefMATHGoogle Scholar
  40. [Sha03a]
    Shapiro, A.: Inference of statistical bounds for multistage stochastic programming problems. Mathematical Methods of Operations Research. 58, 57–68 (2003)MATHMathSciNetCrossRefGoogle Scholar
  41. [Sha03b]
    Shapiro, A.: Monte Carlo sampling methods. In: Rusczyński, A., Shapiro, A. (eds) Stochastic Programming, volume 10 of Handbooks in Operations Research and Management Science. North-Holland (2003)Google Scholar
  42. [Sha04]
    Shapiro, A.: Worst-case distribution analysis of stochastic programs. E-print available at: (2004)Google Scholar
  43. [Sha05a]
    Shapiro, A.: Stochastic programming with equilibrium constraints. Journal of Optimization Theory and Applications (to appear). E-print available at: (2005)Google Scholar
  44. [Sha05b]
    Shapiro, A.: On complexity of multistage stochastic programs. Operations Research Letters (to appear). E-print available at: (2005)Google Scholar
  45. [TA04]
    Takriti, S., Ahmed, S.: On robust optimization of two-stage systems. Mathematical Programming, 99, 109–126 (2004)MathSciNetCrossRefMATHGoogle Scholar
  46. [VAKNS03]
    Verweij, B., Ahmed, S., Kleywegt, A.J., Nemhauser, G., Shapiro, A.: The sample average approximation method applied to stochastic routing problems: a computational study. Computational Optimization and Applications, 24, 289–333 (2003)MathSciNetCrossRefMATHGoogle Scholar
  47. [Val79]
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science, 80, 189–201 (1979)MathSciNetCrossRefGoogle Scholar
  48. [Zac66]
    Žáčková, J.: On minimax solutions of stochastic linear programming problems. Čas. Pěst. Mat., 91, 423–430 (1966)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Alexander Shapiro
    • 1
  • Arkadi Nemirovski
    • 2
  1. 1.Georgia Institute of TechnologyAtlantaUSA
  2. 2.Technion — Israel Institute of TechnologyHaifaIsrael

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