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

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  • Martin Branda

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. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Martin Branda, 2013. "On relations between chance constrained and penalty function problems under discrete distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(2), pages 265-277, April.
    • Handle: RePEc:spr:mathme:v:77:y:2013:i:2:p:265-277
    DOI: 10.1007/s00186-013-0428-7
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    References listed on IDEAS

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    1. Y.M. Ermoliev & T.Y. Ermolieva & G.J. MacDonald & V.I. Norkin, 2000. "Stochastic Optimization of Insurance Portfolios for Managing Exposure to Catastrophic Risks," Annals of Operations Research, Springer, vol. 99(1), pages 207-225, December.
    2. Dupacova, Jitka & Gaivoronski, Alexei & Kos, Zdenek & Szantai, Tamas, 1991. "Stochastic programming in water management: A case study and a comparison of solution techniques," European Journal of Operational Research, Elsevier, vol. 52(1), pages 28-44, May.
    3. William M. Raike, 1970. "Dissection Methods for Solutions in Chance Constrained Programming Problems Under Discrete Distributions," Management Science, INFORMS, vol. 16(11), pages 708-715, July.
    4. Huifu Xu & Dali Zhang, 2012. "Monte Carlo methods for mean-risk optimization and portfolio selection," Computational Management Science, Springer, vol. 9(1), pages 3-29, February.
    5. Martin Branda & Jitka Dupačová, 2012. "Approximation and contamination bounds for probabilistic programs," Annals of Operations Research, Springer, vol. 193(1), pages 3-19, March.
    6. Willem Haneveld & Maarten Vlerk, 2006. "Integrated Chance Constraints: Reduced Forms and an Algorithm," Computational Management Science, Springer, vol. 3(4), pages 245-269, September.
    7. Meskarian, Rudabeh & Xu, Huifu & Fliege, Jörg, 2012. "Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization," European Journal of Operational Research, Elsevier, vol. 216(2), pages 376-385.
    8. Thorsten Koch & Ted Ralphs & Yuji Shinano, 2012. "Could we use a million cores to solve an integer program?," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(1), pages 67-93, August.
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    1. Jiang, J. & Ng, K.M. & Teo, K.M., 2016. "Satisficing measure approach for vehicle routing problem with time windows under uncertaintyAuthor-Name: Nguyen, V.A," European Journal of Operational Research, Elsevier, vol. 248(2), pages 404-414.

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