Geodesic
Chance constrained problems: penalty reformulation and performance of sample approximation technique
Kybernetika, Tome 48 (2012) no. 1, pp. 105-122 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 62A10, 93E12
Keywords: chance constrained problems; penalty functions; asymptotic equivalence; sample approximation technique; investment problem 
@article{KYB_2012_48_1_a5,
     author = {Branda, Martin},
     title = {Chance constrained problems: penalty reformulation and performance of sample approximation technique},
     journal = {Kybernetika},
     pages = {105--122},
     year = {2012},
     volume = {48},
     number = {1},
     mrnumber = {2932930},
     zbl = {1243.93117},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2012_48_1_a5/}
}
TY  - JOUR
AU  - Branda, Martin
TI  - Chance constrained problems: penalty reformulation and performance of sample approximation technique
JO  - Kybernetika
PY  - 2012
SP  - 105
EP  - 122
VL  - 48
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2012_48_1_a5/
LA  - en
ID  - KYB_2012_48_1_a5
ER  - 
%0 Journal Article
%A Branda, Martin
%T Chance constrained problems: penalty reformulation and performance of sample approximation technique
%J Kybernetika
%D 2012
%P 105-122
%V 48
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2012_48_1_a5/
%G en
%F KYB_2012_48_1_a5
Branda, Martin. Chance constrained problems: penalty reformulation and performance of sample approximation technique. Kybernetika, Tome 48 (2012) no. 1, pp. 105-122. http://geodesic.mathdoc.fr/item/KYB_2012_48_1_a5/

[1] S. Ahmed, 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.

[2] E. Angelelli, R. Mansini, M. G. Speranza: A comparison of MAD and CVaR models with real features. J. Banking Finance 32 (2008), 1188-1197. | DOI

[3] M. S. Bazara, H. D. Sherali, C. M. Shetty: Nonlinear Programming: Theory and Algorithms. Wiley, Singapore 1993. | MR

[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.

[5] M. Branda: Stochastic programming problems with generalized integrated chance constraints. Accepted to Optimization 2011. | MR

[6] M. Branda, J. Dupačová: Approximations and contamination bounds for probabilistic programs. Accepted to Ann. Oper. Res. 2011 (Online first). See also SPEPS 13, 2008. | MR

[7] G. Calafiore, M. C. Campi: Uncertain convex programs: randomized solutions and confidence levels. Math. Programming, Ser. A 102 (2008), 25-46. | DOI | MR

[8] A. DasGupta: Asymptotic Theory of Statistics and Probability. Springer, New York 1993. | MR

[9] J. Dupačová, M. Kopa: Robustness in stochastic programs with risk constraints. Accepted to Ann. Oper. Res. 2011 (Online first). | MR

[10] J. Dupačová, A. Gaivoronski, Z. Kos, T. Szantai: Stochastic programming in water management: A case study and a comparison of solution techniques. Europ. J. Oper. Res. 52 (1991), 28-44. | DOI | Zbl

[11] Y. M. Ermoliev, T. Y. Ermolieva, G. J. Macdonald, V. I. Norkin: Stochastic optimization of insurance portfolios for managing exposure to catastrophic risks. Ann. Oper. Res. 99 (2000), 207-225. | DOI | MR | Zbl

[12] P. Lachout: Approximative solutions of stochastic optimization problems. Kybernetika 46 (2010), 3, 513-523. | MR | Zbl

[13] J. Luedtke, S. Ahmed: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19 (2008), 674-699. | DOI | MR | Zbl

[14] J. Nocedal, S. J. Wright: Numerical Optimization. Springer, New York 2000. | MR

[15] B. Pagnoncelli, S. Ahmed, A. Shapiro: Computational study of a chance constrained portfolio selection problem. Optimization Online 2008.

[16] B. Pagnoncelli, S. Ahmed, A. Shapiro: Sample average approximation method for chance constrained programming: Theory and applications. J. Optim. Theory Appl. 142 (2009), 399-416. | DOI | MR | Zbl

[17] A. Prékopa: Contributions to the theory of stochastic programming. Math. Programming 4 (1973), 202-221. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[19] A. Prékopa: Stochastic Programming. Kluwer, Dordrecht and Académiai Kiadó, Budapest 1995. | MR | Zbl

[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. | MR

[21] R. T. Rockafellar, S. Uryasev: Conditional value-at-risk for general loss distributions. J. Banking Finance 26 (2002), 1443-1471. | DOI

[22] R. T. Rockafellar, R. Wets: Variational Analysis. Springer-Verlag, Berlin 2004. | MR

[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. | MR

[24] S. W. Wallace, W. T. Ziemba: Applications of stochastic programming. MPS-SIAM Book Series on Optimization 5 (2005), Society for Industrial and Applied Mathematics. | MR | Zbl

[25] E. Žampachová, 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.

[26] E. Žampachová, P. Popela, M. Mrázek: Optimum beam design via stochastic programming. Kybernetika 46 (2010), 3, 571-582. | MR | Zbl