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Sample average approximation of stochastic dominance constrained programs

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In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with the case of multi-variate stochastic dominance under general distributions and nonlinear functions. We introduce the concept of \({\mathcal{C}}\)-dominance, which generalizes some notions of multi-variate dominance found in the literature. We apply the Sample Average Approximation (SAA) method to this problem, which results in a semi-infinite program, and study asymptotic convergence of optimal values and optimal solutions, as well as the rate of convergence of the feasibility set of the resulting semi-infinite program as the sample size goes to infinity. We develop a finitely convergent method to find an \({\epsilon}\)-optimal solution of the SAA problem. An important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value. We also show that the bounds are asymptotically tight as the sample size goes to infinity.

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References

  1. Aretoulis, G.N., Kalfakakou, G.P., Striagka, F.Z.: Construction material supplier selection under multiple criteria. Oper. Res. (inprint, 2009)

  2. Atlason J., Epelman M., Henderson G.: Call center staffing with simulation and cutting plane methods. Ann. Oper. Res. 127, 333–358 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dembo A., Zeitouni O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998)

    MATH  Google Scholar 

  4. Dentcheva D., Ruszczyński A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14(2), 548–566 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dentcheva D., Ruszczyński A.: Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints. Math. Program. 99, 329–350 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dentcheva D., Ruszczyński A.: Portfolio optimization with stochastic dominance constraints. J. Bank. Fin. 30, 433–451 (2006)

    Article  Google Scholar 

  7. Dentcheva D., Ruszczyński A.: Optimization with multivariate stochastic dominance constraints. Math. Program. 117, 111–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dentcheva D., Henrion R., Ruszczyński A.: Stability and sensitivity of optimization problems with first order stochastic dominance constraints. SIAM J. Optim. 18(1), 322–337 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Drapkin, D., Schultz, R.: An algorithm for stochastic programs with first-order dominance constraints induced by linear recourse. Manuscript, Department of Mathematics, University of Duisburg-Essen, Duisburg, Germany (2007)

  10. Gollmer, R., Gotzes, U., Neise, F., Schultz, R.: Risk modeling via stochastic dominance in power systems with dispersed generation. Manuscript, Department of Mathematics, University of Duisburg-Essen, Duisburg, Germany (2007)

  11. Hiriart-Urruty J., Lemaréchal C.: Convex Analysis and Minimization Algorithms I. Springer, New York (1993)

    Google Scholar 

  12. Homem-de-Mello T., Mehrotra S.: A cutting surface method for linear programs with polyhedral stochastic dominance constraints. SIAM J. Optim. 20(3), 1250–1273 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Horst R., Pardalos P.M., Thoai N.V.: Introduction to Global Optimization. Kluwer, Boston (1995)

    MATH  Google Scholar 

  14. Hu, J., Homem-de-Mello, T., Mehrotra, S.: Multi-criterion robust and stochastic dominance-constrained models with application to budget allocation in homeland security. Manuscript, available at. http://www.optimization-online.org/DB_HTML/2010/04/2605.html (2010)

  15. Karoui N.E., Meziou A.: Constrained optimization with respect to stochastic dominance: Application to portfolio insurance. Math. Fin. 16(1), 103–117 (2006)

    Article  MATH  Google Scholar 

  16. Kleywegt A.J., Shapiro A., Homem-de-Mello T.: The sample average approximation method for stochastic discrete optimiztion. SIAM J. Optim. 12, 479–502 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luedtke J.: New formulations for optimization under stochastic dominance constraints. SIAM J. Optim. 19(3), 1433–1450 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mak W.K., Morton D.P., Wood R.K.: Monte carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24, 47–56 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Müller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Nie, Y., Wu, X., Homem-de-Mello, T.: Optimal path problems with second-order stochastic dominance constraints. Manuscript, Northwestern University (2009)

  22. O’Brien, M.: Techniques for incorporating expected value constraints into stochastic programs. PHD Dissertation, Stanford University (2000)

  23. Prato T., Herath G.: Multiple-criteria decision analysis for integrated catchment management. Ecol. Econ. 63(2–3), 627–632 (2007)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  25. Roman D., Darby-Dowman K., Mitra G.: Portfolio construction based on stochastic dominance and target return distributions. Math. Program. 108, 541–569 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shaked M., Shanthikumar J.G.: Stochastic Orders and their Applications. Academic Press, Boston (1994)

    MATH  Google Scholar 

  27. Shapiro A.: Monte Carlo sampling methods. In: Ruszczynski, A., Shapiro, A. (eds) Handbook of Stochastic Optimization, Elsevier Science Publishers B.V, Amsterdam (2003)

    Google Scholar 

  28. Shapiro A., Dentcheva D., Ruszczyński A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  29. Vogel S.: A stochastic approach to stability in stochastic programming. J. Comput. Appl. Math. 56, 65–96 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. von Neumann J., Morgenstern O.: Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, Princeton (1947)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang K., Trewn J.: Multivariate Statistical Methods in Quality Management. McGraw-Hill, New York (2004)

    Google Scholar 

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Correspondence to Jian Hu.

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Hu, J., Homem-de-Mello, T. & Mehrotra, S. Sample average approximation of stochastic dominance constrained programs. Math. Program. 133, 171–201 (2012). https://doi.org/10.1007/s10107-010-0428-9

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