Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

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.

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

  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)

    Article  Google Scholar 

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

    Google Scholar 

  3. Dupačová, 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)

    Article  MATH  Google Scholar 

  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)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil, 22453-900

    B. K. Pagnoncelli

  2. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    S. Ahmed & A. Shapiro

Authors
  1. B. K. Pagnoncelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. K. Pagnoncelli.

Additional information

Communicated by P.M. Pardalos.

B.K. Pagnoncelli’s research was supported by CAPES and FUNENSEG.

S. Ahmed’s research was partly supported by the NSF Award DMI-0133943.

A. Shapiro’s research was partly supported by the NSF Award DMI-0619977.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pagnoncelli, B.K., Ahmed, S. & Shapiro, A. Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications. J Optim Theory Appl 142, 399–416 (2009). https://doi.org/10.1007/s10957-009-9523-6

Download citation

  • Published:

  • Issue Date:

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

Keywords

Search

Navigation