Mathematical Methods of Operations Research

, Volume 65, Issue 1, pp 115–140 | Cite as

On two-stage convex chance constrained problems

Original Article


In this paper we develop approximation algorithms for two-stage convex chance constrained problems. Nemirovski and Shapiro (Probab Randomized Methods Des Uncertain 2004) formulated this class of problems and proposed an ellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine. We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm (Nemirovski and Shapiro in Probab Randomized Methods Des Uncertain 2004). Since the polytope determining r is random, computing r is difficult. Yet, the solution algorithm requires r as an input. In this paper we provide some guidance for selecting r. We show that the largest value of r is determined by the degree of robust feasibility of the two-stage chance constrained problem—the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous two-stage chance constrained problems. In this formulation, the random variables defining the chance constraint are known to have a fixed distribution; however, the decision maker is only able to estimate this distribution to within some error. We construct an algorithm that solves the ambiguous two-stage chance constrained problem when the impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope are known explicitly.


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  1. Anthony M, Biggs N (1992) Computational learning theory. Cambridge University Press, CambridgeMATHGoogle Scholar
  2. Atamtürk A, Zhang M (2005) Two-stage robust network flow and design for demand uncertainty. ManuscriptGoogle Scholar
  3. Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2, Ser. A):351–376MATHCrossRefMathSciNetGoogle Scholar
  4. Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23(4):769–805MATHMathSciNetCrossRefGoogle Scholar
  5. Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization. SIAM, PhiladelphiaMATHGoogle Scholar
  6. Bertsimas D, Sim M (2004) Robust conic optimization. Under review in Math ProgGoogle Scholar
  7. Bukszár J (2001) Upper bounds for the probability of a union by multitrees. Adv Appl Probab 33:437–452MATHCrossRefGoogle Scholar
  8. Calafiore G, Campi MC (2003) Uncertain convex programs: Randomized solutions and confidence levels. To appear in Math ProgGoogle Scholar
  9. Calafiore G, Campi MC (2004) Decision making in an uncertain environment: the scenario-based optimization approach. Working paperGoogle Scholar
  10. de Farias DP, Van Roy B (2001) On constraint sampling in the linear programming approach to approximate dynamic programming. To appear in Math Oper ResGoogle Scholar
  11. Dentcheva D, Prékopa A, Ruszczyński A (2000) Concavity and efficient points of discrete distributions in probabilistic programming. Math Program 89(1, Ser. A):55–77MATHCrossRefMathSciNetGoogle Scholar
  12. Dupačová J (2001) Stochastic programming: minimax approach. In Encyclopedia of Optimization. KluwerGoogle Scholar
  13. Erdoğan E, Iyengar G (2004) Ambiguous chance constrained problems and robust optimization. To appear in Math ProgGoogle Scholar
  14. Erdoğan E, Iyengar G (2005) Boosting, importance sampling, and convex chance constrained problem. ManuscriptGoogle Scholar
  15. Freund RM, Vera JR (1999) Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm. SIAM J Opt 10:155–176MATHCrossRefMathSciNetGoogle Scholar
  16. Goldfarb D, Iyengar G (2003a) Robust convex quadratically constrained programs. Math Program Ser B. 97(3):495–515MATHCrossRefMathSciNetGoogle Scholar
  17. Goldfarb D, Iyengar G (2003b) Robust portfolio selection problems. Math Oper Res 28(1):1–38MATHCrossRefMathSciNetGoogle Scholar
  18. Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, LondonMATHGoogle Scholar
  19. Henrion R (2005) Structural properties of linear probabilistic constraints. Stochastic programming E-print series (SPEPS), 13Google Scholar
  20. Kan Yuri (2002) Application of the quantile optimization to bond portfolio selection. In: Stochastic optimization techniques (Neubiberg/Munich, 2000), vol 513 of Lecture Notes in Econom and Math Systems, pp 285–308. Springer, Berlin Heidelberg New YorkGoogle Scholar
  21. Kearns MJ, Vazirani UV (1997) An introduction to computational learning theory. MIT Press, CambridgeGoogle Scholar
  22. Khachiyan LG (1979) A polynomial algorithm in linear programming. Doklady Akademiia Nauk SSSR, 244(S):1093–1096 Translated in Soviet Mathematics Doklady 20:1Google Scholar
  23. Lagoa CM, Li X, Sznaier M (2005) Probabilistically constrained linear programs and risk-adjusted controller design. SIAM J Optim 15:938–951MATHCrossRefMathSciNetGoogle Scholar
  24. Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second-order cone programming. Linear Algebra Appl 284(1-3):193–228MATHCrossRefMathSciNetGoogle Scholar
  25. Nemirovski A (2003) On tractable approximations of randomly perturbed convex constraints. In Proc. 42nd IEEE Conf. Dec. Contr. (CDC), vol 3, pp 2419–2422Google Scholar
  26. Nemirovski A, Shapiro A (2004) Scenario approximations of chance constraints. To appear in Probab Randomized Methods Des UncertainGoogle Scholar
  27. Prekopa A (1995) Stochastic Programming. Kluwer, DordrechtGoogle Scholar
  28. Rachev ST (1991) Probability metrics and the stability of stochastic models. Wiley, LondonMATHGoogle Scholar
  29. Renegar J (1994) Some perturbation theory for linear programming. Math Prog 65:73–91CrossRefMathSciNetGoogle Scholar
  30. Renegar J (1995) Linear programming, complexity theory and elementary functional analysis. Math Prog 70:279–351MathSciNetGoogle Scholar
  31. Ruszczynski A, Shapiro A (eds) (2003) Stochastic programming. Handbook in Operations Research and Management Science. Elsevier, AmsterdamGoogle Scholar
  32. Shapiro A Some recent developments in stochastic programming. ORB Newsletter, Available at 13, March 2004Google Scholar
  33. Shapiro A, Ahmed S (2004) On a class of minimax stochastic programs. To appear in SIAM J OptGoogle Scholar
  34. Shapiro A, Kleywegt AJ (2002) Minimax analysis of stochastic problems. Optim Methods Softw 17:523–542MATHCrossRefMathSciNetGoogle Scholar
  35. Shor NZ (1977) Cut-off method with space extension in convex programming problems. Cybern 13:94–96Google Scholar
  36. Vapnik VN (1995) The nature of statistical learning theory. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  37. Žáčková J (1966) On minimax solutions of stochastic linear programs. Čas Pěst Mat 423–430Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.IEOR DepartmentColumbia UniversityNew YorkUSA

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