Kybernetika 48 no. 1, 105-122, 2012

Chance constrained problems: penalty reformulation and performance of sample approximation technique

Martin Branda


We explore reformulation of nonlinear stochastic programs with several joint chance constraints by stochastic programs with suitably chosen penalty-type objectives. We show that the two problems are asymptotically equivalent. Simpler cases with one chance constraint and particular penalty functions were studied in [6,11]. The obtained problems with penalties and with a fixed set of feasible solutions are simpler to solve and analyze then the chance constrained programs. We discuss solving both problems using Monte-Carlo simulation techniques for the cases when the set of feasible solution is finite or infinite bounded. The approach is applied to a financial optimization problem with Value at Risk constraint, transaction costs and integer allocations. We compare the ability to generate a feasible solution of the original chance constrained problem using the sample approximations of the chance constraints directly or via sample approximation of the penalty function objective.


chance constrained problems, penalty functions, asymptotic equivalence, sample approximation technique, investment problem


93E12, 62A10


  1. S. Ahmed and A. Shapiro: Solving chance-constrained stochastic programs via sampling and integer programming. In: Tutorials in Operations Research, (Z.-L. Chen and S. Raghavan, eds.), INFORMS 2008.   CrossRef
  2. E. Angelelli, R. Mansini and M. G. Speranza: A comparison of MAD and CVaR models with real features. J. Banking Finance 32 (2008), 1188-1197.   CrossRef
  3. M. S. Bazara, H. D. Sherali and C. M. Shetty: Nonlinear Programming: Theory and Algorithms. Wiley, Singapore 1993.   CrossRef
  4. M. Branda: Solving real-life portfolio problem using stochastic programming and Monte-Carlo techniques. In: Proc. Mathematical Methods in Economics 2010, (M. Houda, J. Friebelová, eds.), University of South Bohemia, České Budějovice 2010.   CrossRef
  5. M. Branda: Stochastic programming problems with generalized integrated chance constraints. Accepted to Optimization 2011.   CrossRef
  6. M. Branda and J. Dupačová: Approximations and contamination bounds for probabilistic programs. Accepted to Ann. Oper. Res. 2011 (Online first). See also SPEPS 13, 2008.   CrossRef
  7. G. Calafiore and M. C. Campi: Uncertain convex programs: randomized solutions and confidence levels. Math. Programming, Ser. A 102 (2008), 25-46.   CrossRef
  8. A. DasGupta: Asymptotic Theory of Statistics and Probability. Springer, New York 1993.   CrossRef
  9. J. Dupačová and M. Kopa: Robustness in stochastic programs with risk constraints. Accepted to Ann. Oper. Res. 2011 (Online first).   CrossRef
  10. J. Dupačová, A. Gaivoronski, Z. Kos and T. Szantai: Stochastic programming in water management: A case study and a comparison of solution techniques. Europ. J. Oper. Res. 52 (1991), 28-44.   CrossRef
  11. Y. M. Ermoliev, T. Y. Ermolieva, G. J. Macdonald and V. I. Norkin: Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Ann. Oper. Res. 99 (2000), 207-225.   CrossRef
  12. P. Lachout: Approximative solutions of stochastic optimization problems. Kybernetika 46 (2010), 3, 513-523.   CrossRef
  13. J. Luedtke and S. Ahmed: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19 (2008), 674-699.   CrossRef
  14. J. Nocedal and S. J. Wright: Numerical Optimization. Springer, New York 2000.   CrossRef
  15. B. Pagnoncelli, S. Ahmed and A. Shapiro: Computational study of a chance constrained portfolio selection problem. Optimization Online 2008.   CrossRef
  16. B. Pagnoncelli, S. Ahmed and A. Shapiro: Sample average approximation method for chance constrained programming: Theory and applications. J. Optim. Theory Appl. {\mi 142} (2009), 399-416.   CrossRef
  17. A. Prékopa: Contributions to the theory of stochastic programming. Math. Programming 4 (1973), 202-221.   CrossRef
  18. A. Prékopa: Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution. Math. Methods Oper. Res. 34 (1990), 441-461.   CrossRef
  19. A. Prékopa: Stochastic Programming. Kluwer, Dordrecht and Académiai Kiadó, Budapest 1995.   CrossRef
  20. A. Prékopa: Probabilistic programming. In: Stochastic Programming, (A. Ruszczynski and A. Shapiro,eds.), Handbook in Operations Research and Management Science, Vol. 10, Elsevier, Amsterdam 2003, pp. 267-352.   CrossRef
  21. R. T. Rockafellar and S. Uryasev: Conditional value-at-risk for general loss distributions. J. Banking Finance 26 (2002), 1443-1471.   CrossRef
  22. R. T. Rockafellar and R. Wets: Variational Analysis. Springer-Verlag, Berlin 2004.   CrossRef
  23. A. Shapiro: Monte Carlo sampling methods. In: Stochastic Programming, (A. Ruszczynski and A. Shapiro, eds.), Handbook in Operations Research and Management Science, Vol. 10, Elsevier, Amsterdam 2003, pp. 353-426.   CrossRef
  24. S. W. Wallace and W. T. Ziemba: Applications of stochastic programming. MPS-SIAM Book Series on Optimization 5 (2005), Society for Industrial and Applied Mathematics.   CrossRef
  25. E. Žampachová and M. Mrázek: Stochastic optimization in beam design and its reliability check. In: MENDEL 2010 - 16th Internat. Conference on Soft Computing, (R. Matoušek), ed.), Mendel Journal series, FME BUT, Brno 2010, pp. 405-410.   CrossRef
  26. E. Žampachová, P. Popela and M. Mrázek: Optimum beam design via stochastic programming. Kybernetika 46 (2010), 3, 571-582.   CrossRef