Advertisement

Mathematical Programming

, Volume 107, Issue 1–2, pp 37–61 | Cite as

Ambiguous chance constrained problems and robust optimization

  • E. Erdoğan
  • G. IyengarEmail author
Article

Abstract

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set Open image in new window of the distributions is of the form Open image in new window where ρ p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Open image in new window The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice.

Keywords

Robust optimization Stochastic programming Learning Theory Coupling of random variables Ambiguity in measure Sample approximation VC dimension 

Mathematics Subject Classification (2000)

62G35 90C15 68Q32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anthony, M., Biggs, N.: Computational Learning Theory. Cambridge University Press, 1992Google Scholar
  2. 2.
    Artzner, P., Delbaen, F., Eber, J.P., Heath, D.: Coherent risk measures. Math. Finance 9, 203–228 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ben-Tal, A., Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim. 7 (4), 991–1016 (1997)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23 (4), 769–805 (1998)MathSciNetGoogle Scholar
  5. 5.
    Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25 (1), 1–13 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. SIAM, Philadelphia, PA, 2001Google Scholar
  7. 7.
    Bertsimas, D., Sim, M.: Robust conic optimization (May 2004). Under review in Math. Prog.Google Scholar
  8. 8.
    Birge, J.R., Wets, J.R.B.: Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse. Math. Prog. Study 27, 54–102 (1986)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Learnability and the Vapnik-Chervonenkis dimension. J. ACM 36 (4), 929–965 (1989)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Calafiore, G., Campi, M.C.: Uncertain convex programs: Randomized solutions and confidence levels, 2003. To appear in Math. Prog.Google Scholar
  11. 11.
    Calafiore, G., Campi, M.C.: Decision making in an uncertain environment: the scenario-based optimization approach, 2004. Working paperGoogle Scholar
  12. 12.
    Charnes, A., Cooper, W.W.: Uncertain convex programs: randomized solutions and confidence levels. Mgmt. Sc. 6, 73–79 (1959)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Chen, Z., Epstein, L.: Ambiguity, risk and asset returns in continuous time, 2000. MimeoGoogle Scholar
  14. 14.
    Dudley, R.M.: Distance of probability measures and random variables. Ann. Math. Stat. 39, 1563–1572 (1968)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Dupačová, J.: The minimax approach to stochastic program and illustrative application. Stochastics 20, 73–88 (1987)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dupačová, J.: Stochastic programming: minimax approach. In: Encyclopedia of Optimization. Kluwer, 2001Google Scholar
  17. 17.
    El Ghaoui, L., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18 (4), 1035–1064 (1997)CrossRefGoogle Scholar
  18. 18.
    El Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9 (1), 33–52 (1998)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Epstein, L.G., Schneider, M.: Recursive multiple priors. Tech. Rep. 485, Rochester Center for Economic Research, 2001. Available at http://rcer.econ.rochester.edu. To appear in J. Econ. Theory Google Scholar
  20. 20.
    Epstein, L.G., Schneider, M.: Learning under Ambiguity. Tech. Rep. 497, Rochester Center for Economic Research, 2002. Available at http://rcer.econ.rochester.edu.Google Scholar
  21. 21.
    Erdoğan, E., Iyengar, G.: Approximation algorithms for ambiguous chance constrained problems. February, 2004Google Scholar
  22. 22.
    de Farias, D.P., Van Roy, B.: On constraint sampling in the linear programming approach to approximate dynamic programming, 2001. To appear in Math. Oper. Res.Google Scholar
  23. 23.
    Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Fin. and Stoch. 6, 429–447 (2002)CrossRefzbMATHGoogle Scholar
  24. 24.
    El Ghaoui, L., Nilim, A.: The curse of uncertainty in dynamic programming and how to fix it, 2002. To appear in Oper. Res.. UC Berkeley Tech Report UCB-ERL-M02/31Google Scholar
  25. 25.
    Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Intl. Stat. Rev. 7 (3), 419–435 (2002)Google Scholar
  26. 26.
    Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18 (2), 141–153 (1989)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Goldfarb, D., Iyengar, G.: Robust portfolio selection problems. Math. Oper. Res. 28 (1), 1–38 (2003)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Hansen, L.P., Sargent, T.J.: Robust control and model uncertainty. American Economic Review 91, 60–66 (2001)CrossRefGoogle Scholar
  29. 29.
    Iyengar, G.: Robust dynamic programming 2002. To appear in Math. Oper. Res.. Available at http://www.corc.ieor.columbia.edu/reports/techreports/tr-2002-07.pd f
  30. 30.
    Jagannathan, R.: Minimax procedure for a class of linear programs under uncertainty. Oper. Res. 25, 173–177 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Kearns, M.J., Vazirani, U.V.: An introduction to computational learning theory. MIT Press, Cambridge, MA, 1997Google Scholar
  32. 32.
    Long, P.M.: The complexity of learning according to two models of a drifting environment. Machine Learning 37 (3), 337–354 (1999)CrossRefGoogle Scholar
  33. 33.
    Nemirovski, A.: On tractable approximations of randomly perturbed convex constraints. In: Proc. 42nd IEEE Conf. Dec. Contr. (CDC), vol. 3, 2003, pp. 2419–2422Google Scholar
  34. 34.
    Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints, 2004. To appear in Probabilistic and randomized methods for design under uncertaintyGoogle Scholar
  35. 35.
    Rachev, S.T.: Probability metrics and the stability of stochastic models. John Wiley & Sons, 1991Google Scholar
  36. 36.
    Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton, New Jersey, 1997Google Scholar
  37. 37.
    Ruszczynski, A., Shapiro, A. (eds.): Stochastic Programming. Handbook in Operations Research and Management Science. Elsevier, 2003Google Scholar
  38. 38.
    Ruszczynski, A., Shapiro, A.: Optimization of risk measures (2004). Available at http://www.optimization-online.org/DB_HTML/2004/02/822.html
  39. 39.
    Ruszczynski, A., Shapiro, A.: Optimization of risk measures (2004). Available at http://ideas.repec. org/p/wpa/wuwpri/0407002.htmlGoogle Scholar
  40. 40.
    Shapiro, A.: Some recent developments in stochastic programming. ORB Newsletter 13, 2004. Available at http://www.ballarat.edu.au/ard/itms/CIAO/ORBNewsletter/issue13. shtml#11
  41. 41.
    Shapiro, A., Ahmed, S.: On a class of minimax stochastic programs, 2004. To appear in SIAM J. Opt.Google Scholar
  42. 42.
    Shapiro, A., Kleywegt, A.J.: Minimax analysis of stochastic problems. Optimization Methods and Software 17, 523–542 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Thorisson, H.: Coupling, Stationary, and Regeneration. Probability and its Applications. Springer-Verlag, 2000Google Scholar
  45. 45.
    Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer, New York, NY, 1995Google Scholar
  46. 46.
    Žáčková, J.: On minimax solutions of stochastic linear programs. Čas. Pěst. Mat. 1966, pp. 423–430Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.IEOR DepartmentColumbia UniversityNew YorkUSA

Personalised recommendations