Skip to main content
SpringerLink
Log in

On relations between chance constrained and penalty function problems under discrete distributions

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

We extend the theory of penalty functions to stochastic programming problems with nonlinear inequality constraints dependent on a random vector with known distribution. We show that the problems with penalty objective, penalty constraints and chance constraints are asymptotically equivalent under discretely distributed random parts. This is a complementary result to Branda (Kybernetika 48(1):105–122, 2012a), Branda and Dupačová (Ann Oper Res 193(1):3–19, 2012), and Ermoliev et al. (Ann Oper Res 99:207–225, 2000) where the theorems were restricted to continuous distributions only. We propose bounds on optimal values and convergence of optimal solutions. Moreover, we apply exact penalization under modified calmness property to improve the results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms, 3rd edn. Wiley, Singapore

    Book  Google Scholar 

  • Branda M (2010) Solving real-life portfolio problem using stochastic programming and Monte-Carlo techniques. In: Houda M, Friebelová J (eds) Proceedings of the 28th international conference on mathematical methods in economics 2010. University of South Bohemia, České Budějovice, pp 67–72

  • Branda M (2012a) Chance constrained problems: penalty reformulation and performance of sample approximation technique. Kybernetika 48(1):105–122

    Google Scholar 

  • Branda M (2012b) Sample approximation technique for mixed-integer stochastic programming problems with several chance constraints. Oper Res Lett 40(3):207–211

    Google Scholar 

  • Branda M (2012c) Stochastic programming problems with generalized integrated chance constraints. Optim J Math Program Oper Res 61(3):949–968

    Google Scholar 

  • Branda M, Dupačová J (2012) Approximations and contamination bounds for probabilistic programs. Ann Oper Res 193(1):3–19

    Article  MathSciNet  MATH  Google Scholar 

  • Burke JV (1991a) Calmness and exact penalization. SIAM J Control Optim 29:493–497

    Google Scholar 

  • Burke JV (1991b) An exact penalization viewpoint of constrained optimization. SIAM J Control Optim 29:968–998

    Google Scholar 

  • D’Ambrosio C, Lodi A (2011) Mixed integer nonlinear programming tools: a practical overview. 4OR Q J Oper Res 9:329–349

    Article  MathSciNet  MATH  Google Scholar 

  • Clarke FH (1983) Optimization and nonsmooth analysis. Wiley, New York

    MATH  Google Scholar 

  • Dupačová J, Gaivoronski A, Kos Z, Szantai T (1991) Stochastic programming in water management: a case study and a comparison of solution techniques. Eur J Oper Res 52:28–44

    Article  MATH  Google Scholar 

  • Ermoliev YM, Ermolieva TY, Macdonald GJ, Norkin VI (2000) Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Ann Oper Res 99:207–225

    Article  MathSciNet  MATH  Google Scholar 

  • Klein Haneveld WK (1986) Duality in stochastic linear and dynamic programming. Lecture notes in economics and mathematical systems, vol 274. Springer, New York

  • Klein Haneveld WK, van der Vlerk M (2006) Integrated chance constraints: reduced forms and an algorithm. Comput Manag Sci 3(4):245–269

    Article  MathSciNet  MATH  Google Scholar 

  • Hoheisel T, Kanzowa Ch, Outrata JV (2010) Exact penalty results for mathematical programs with vanishing constraints. Nonlinear Anal 72:2514–2526

    Article  MathSciNet  MATH  Google Scholar 

  • Koch T, Ralphs T, Shinano Y (2012) Could we use a million cores to solve an integer program? Math Methods Oper Res 76(1):67–93

    Article  MathSciNet  MATH  Google Scholar 

  • Luedtke J, Ahmed S (2008) A sample approximation approach for optimization with probabilistic constraints. SIAM J Optim 19:674–699

    Article  MathSciNet  MATH  Google Scholar 

  • Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Meskarian R, Xu H, Fliege J (2012) Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization. Eur J Oper Res 216:376–385

    Article  MathSciNet  MATH  Google Scholar 

  • Prékopa A (1995) Stochastic programming. Kluwer Dordrecht and Académiai Kiad, Budapest

    Book  Google Scholar 

  • Prékopa A (2003) Probabilistic programming. In: Ruszczynski A, Shapiro A (eds) Stochastic programming handbook in operations research and management science, vol 10. Elsevier, Amsterdam, pp 483–554

  • Raike WM (1970) Dissection methods for solutions in chance constrained programming problems under discrete distributions. Manag Sci 16(11):708–715

    Article  MathSciNet  MATH  Google Scholar 

  • Rockafellar RT, Wets R (2004) Variational analysis, 2nd edn. Springer, Berlin

    Google Scholar 

  • Shapiro A (2003) Monte Carlo sampling methods. In: Ruszczynski A, Shapiro A (eds) Stochastic programming, handbook in operations research and management science, vol 10. Elsevier, Amsterdam, pp 483–554

  • Xu H, Zhang D (2012) Monte Carlo methods for mean-risk optimization and portfolio selection. Comput Manag Sci 9(1):3–29

    Article  MathSciNet  Google Scholar 

  • Wang W, Ahmed S (2008) Sample average approximation of expected value constrained stochastic programs. Oper Res Lett 36(5):515–519

    Article  MathSciNet  MATH  Google Scholar 

  • Žampachová E, Mrázek M (2010) Stochastic optimization in beam design and its reliability check. In: MENDEL 2010—16th international conference on soft computing. Mendel Journal series. Brno, FME BUT, pp 405–410

Download references

Acknowledgments

The research was supported by the Czech Science Foundation under the Grant (P402/12/G097).

Author information

Authors and Affiliations

  1. Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Prague, Czech Republic

    Martin Branda

Authors
  1. Martin Branda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Branda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Branda, M. On relations between chance constrained and penalty function problems under discrete distributions. Math Meth Oper Res 77, 265–277 (2013). https://doi.org/10.1007/s00186-013-0428-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-013-0428-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation