article

Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications

Authors:

B. K. Pagnoncelli,

A. ShapiroAuthors Info & Claims

Journal of Optimization Theory and Applications, Volume 142, Issue 2

Pages 399 - 416

https://doi.org/10.1007/s10957-009-9523-6

Published: 01 August 2009 Publication History

Abstract

We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained problems. Numerical experiments are performed to correctly tune the parameters involved in the SAA. In addition, we present a method for constructing statistical lower bounds for the optimal value of the considered problem and discuss how one should tune the underlying parameters. We apply the SAA to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem.

References

[1]

Charnes, A., Cooper, W.W., Symmonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4, 235---263 (1958)

Digital Library

[2]

Prékopa, A.: Stochastic Programming. Kluwer Academic, Dordrecht (1995)

[3]

Dupaă?ová, J., Gaivoronski, A., Kos, Z., Szántai, T.: Stochastic programming in water management: a case study and a comparison of solution techniques. Eur. J. Oper. Res. 52, 28---44 (1991)

[4]

Henrion, R., Li, P., Möller, A., Steinbach, S.G., Wendt, M., Wozny, G.: Stochastic optimization for operating chemical processes under uncertainty. In: Grötschel, M., Krunke, S., Rambau, J. (eds.) Online Optimization of Large Scale Systems, pp. 457---478. Springer, Berlin (2001)

[5]

Henrion, R., Möller, A.: Optimization of a continuous distillation process under random inflow rate. Comput. Math. Appl. 45, 247---262 (2003)

[6]

Dentcheva, D., Prékopa, A., Ruszczyński, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55---77 (2000)

[7]

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

Digital Library

[8]

Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969---996 (2006)

Digital Library

[9]

Shapiro, A.: Monte Carlo sampling methods. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in OR & MS, vol. 10, pp. 353---425. North-Holland, Amsterdam (2003)

[10]

Atlason, J., Epelman, M.A., Henderson, S.G.: Optimizing call center staffing using simulation and analytic center cutting plane methods. Manag. Sci. 54, 295---309 (2008)

Digital Library

[11]

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

Digital Library

[12]

Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

[13]

Artstein, Z., Wets, R.J.-B.: Consistency of minimizers and the SLLN for stochastic programs. J. Convex Anal. 2, 1---17 (1996)

[14]

Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum observations. Ann. Math. Stat. 23, 493---507 (1952)

[15]

Markowitz, H.: Portfolio selection. J. Financ. 7, 77---97 (1952)

[16]

Wang, Y., Chen, Z., Zhang, K.: A chance-constrained portfolio selection problem under t-distribution. Asia Pac. J. Oper. Res. 24(4), 535---556 (2007)

[17]

Campi, M.C., Garatti, S.: The exact feasibility of randomized solutions of robust convex programs. Optimization online (www.optimization-online.org) (2007)

[18]

Calafiore, G., Campi, M.C.: The scenario approach to robust control design. IEEE Trans. Autom. Control. 51, 742---753 (2006)

[19]

Klein Haneveld, W.K., van der Vlerk, M.H.: Stochastic Programming (lecture notes) (2007)

[20]

Law, A., Kelton, W.D.: Simulation Modeling and Analysis. Industrial Engineering and Management Science Series. McGraw---Hill Science/Engineering/Math, New York (1999)

Digital Library

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Published In

cover image Journal of Optimization Theory and Applications

Journal of Optimization Theory and Applications Volume 142, Issue 2

August 2009

158 pages

ISSN:0022-3239

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Copyright © Copyright © 2009 Springer Science+Business Media, LLC.

Publisher

Plenum Press

United States

Publication History

Published: 01 August 2009

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