Mathematical Programming

, Volume 171, Issue 1–2, pp 217–282 | Cite as

Robust sample average approximation

  • Dimitris Bertsimas
  • Vishal Gupta
  • Nathan KallusEmail author
Full Length Paper Series A


Sample average approximation (SAA) is a widely popular approach to data-driven decision-making under uncertainty. Under mild assumptions, SAA is both tractable and enjoys strong asymptotic performance guarantees. Similar guarantees, however, do not typically hold in finite samples. In this paper, we propose a modification of SAA, which we term Robust SAA, which retains SAA’s tractability and asymptotic properties and, additionally, enjoys strong finite-sample performance guarantees. The key to our method is linking SAA, distributionally robust optimization, and hypothesis testing of goodness-of-fit. Beyond Robust SAA, this connection provides a unified perspective enabling us to characterize the finite sample and asymptotic guarantees of various other data-driven procedures that are based upon distributionally robust optimization. This analysis provides insight into the practical performance of these various methods in real applications. We present examples from inventory management and portfolio allocation, and demonstrate numerically that our approach outperforms other data-driven approaches in these applications.


Sample average approximation of stochastic optimization Data-driven optimization Goodness-of-fit testing Distributionally robust optimization Conic programming Inventory management Portfolio allocation 

Mathematics Subject Classification

90C15 62G10 90C47 90C34 90C25 



The authors would like to thank the anonymous reviewers and associate editor for their extremely helpful suggestions and very thorough review of the paper. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.


  1. 1.
    Bassamboo, A., Zeevi, A.: On a data-driven method for staffing large call centers. Oper. Res. 57(3), 714–726 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bayraksan, G., Love, D.K.: Data-driven stochastic programming using phi-divergences. In: Tutorials in Operations Research, pp. 1–19 (2015)Google Scholar
  3. 3.
    Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., Rennen, G.: Robust solutions of optimization problems affected by uncertain probabilities. Manag. Sci. 59(2), 341–357 (2013)CrossRefGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization. In: Society for Industrial and Applied Mathematics. SIAM, Philadelphia (2001)Google Scholar
  5. 5.
    Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  6. 6.
    Bertsimas, D., Doan, X.V., Natarajan, K., Teo, C.P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bertsimas, D., Gupta, V., Kallus, N.: Data-driven robust optimization. Preprint arXiv:1401.0212 (2013)
  8. 8.
    Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  11. 11.
    Birge, J.R., Wets, R.J.B.: Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse. Math. Program. Study 27, 54–102 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Calafiore, G.C., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1), 1–22 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D’Agostino, R.B., Stephens, M.A.: Goodness-of-Fit Techniques. Dekker, New York (1986)zbMATHGoogle Scholar
  15. 15.
    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 55(3), 98–112 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    DeMiguel, V., Garlappi, L., Nogales, F.J., Uppal, R.: A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Manag. Sci. 55(5), 798–812 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Dodge, Y.: The Oxford Dictionary of Statistical Terms. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  18. 18.
    Dudley, R.M.: Real Analysis and Probability, vol. 74. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    Dupačová, J.: The minimax approach to stochastic programming and an illustrative application. Stoch. Int. J. Probab. Stoch. Process. 20(1), 73–88 (1987)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Efron, B.: An Introduction to the Bootstrap. Chapman & Hall, New York (1993)CrossRefzbMATHGoogle Scholar
  21. 21.
    Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning. Springer, New York (2001)zbMATHGoogle Scholar
  22. 22.
    Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)CrossRefzbMATHGoogle Scholar
  23. 23.
    Grotschel, M., Lovasz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  24. 24.
    Gurobi Optimization Inc.: Gurobi optimizer reference manual. (2013)
  25. 25.
    Hoerl, A.E., Kennard, R.W.: Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1), 55–67 (1970)CrossRefzbMATHGoogle Scholar
  26. 26.
    Homem-de Mello, T., Bayraksan, G.: Monte Carlo sampling-based methods for stochastic optimization. Surv. Oper. Res. Manag. Sci. 19(1), 56–85 (2014)MathSciNetGoogle Scholar
  27. 27.
    Jiang, R., Guan, Y.: Data-driven chance constrained stochastic program. Tech. rep., Technical report, University of Florida. Available at: Optimization Online (2013)
  28. 28.
    King, A.J., Wets, R.J.B.: Epiconsistency of convex stochastic programs. Stoch. Stoch. Rep. 34(1–2), 83–92 (1991)CrossRefzbMATHGoogle Scholar
  29. 29.
    Klabjan, D., Simchi-Levi, D., Song, M.: Robust stochastic lot-sizing by means of histograms. Prod. Oper. Manag. 22(3), 691–710 (2013)CrossRefGoogle Scholar
  30. 30.
    Kleywegt, A.J., Shapiro, A., Homem-de Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2), 479–502 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kullback, S.: A lower bound for discrimination information in terms of variation. IEEE Trans. Inf. Theory 13(1), 126–127 (1967)CrossRefGoogle Scholar
  32. 32.
    Levi, R., Perakis, G., Uichanco, J.: The data-driven newsvendor problem: new bounds and insights. Oper. Res. (2015). doi: 10.1287/opre.2015.1422 MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lim, A.E., Shanthikumar, J.G., Vahn, G.Y.: Conditional value-at-risk in portfolio optimization: coherent but fragile. Oper. Res. Lett. 39(3), 163–171 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1), 193–228 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mak, W.K., Morton, D.P., Wood, R.K.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24(1), 47–56 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Noether, G.E.: Note on the Kolmogorov statistic in the discrete case. Metrika 7(1), 115–116 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Popescu, I.: Robust mean-covariance solutions for stochastic optimization. Oper. Res. 55(1), 98–112 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Prékopa, A.: Stochastic Programming. Kluwer Academic Publishers, Dordrecht (1995)CrossRefzbMATHGoogle Scholar
  39. 39.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press, New York (1981)zbMATHGoogle Scholar
  40. 40.
    Rice, J.: Mathematical Statistics and Data Analysis. Thomson/Brooks/Cole, Belmont (2007)Google Scholar
  41. 41.
    Rockafellar, T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  42. 42.
    Rohlf, F.J., Sokal, R.R.: Statistical Tables, 4th edn. Macmillan, New York (2012)zbMATHGoogle Scholar
  43. 43.
    Scarf, H.: A min–max solution of an inventory problem. In: Arrow, K.J., Karlin, S., Scarf, H. (eds.) Studies in the Mathematical Theory of Inventory and Production, pp. 201–209. Stanford University Press, Stanford (1958)Google Scholar
  44. 44.
    Scarsini, M.: Multivariate convex orderings, dependence, and stochastic equality. J. Appl. Probab. 35(1), 93–103 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  46. 46.
    Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.A., López, M.A. (eds.) Semi-Infinite Programming: Recent Advances, pp. 135–165. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  47. 47.
    Shapiro, A., Ruszczyński, A. (eds.): Handbooks in Operations Research and Management Science: Vol. 10. Stochastic Programming. Elsevier, Amsterdam (2003)Google Scholar
  48. 48.
    Shawe-Taylor, J., Cristianini, N.: Estimating the moments of a random vector with applications (2003).
  49. 49.
    Stephens, M.A.: Use of the Kolmogorov–Smirnov, Cramér–Von Mises and related statistics without extensive tables. J. R. Stat. Soc. B 32(1), 115–122 (1970)zbMATHGoogle Scholar
  50. 50.
    Thas, O.: Comparing Distributions. Springer, New York (2009)zbMATHGoogle Scholar
  51. 51.
    Tikhonov, A.N.: On the stability of inverse problems. Dokl. Akad. Nauk SSSR 39(5), 195–198 (1943)MathSciNetGoogle Scholar
  52. 52.
    Vapnik, V.: Principles of risk minimization for learning theory. In: NIPS, pp. 831–838 (1991)Google Scholar
  53. 53.
    Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)zbMATHGoogle Scholar
  54. 54.
    Wächter, A., Biegler, L.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Wang, Z., Glynn, P., Ye, Y.: Likelihood robust optimization for data-driven problems. Preprint arXiv:1307.6279 (2013)
  56. 56.
    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Oper. Res. 62(6), 1358–1376 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Žáčková, J.: On minimax solutions of stochastic linear programming problems. Časopis pro pěstování matematiky 91(4), 423–430 (1966)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Sloan School of ManagementMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Marshall School of BusinessUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.School of Operations Research and Information Engineering and Cornell TechCornell UniversityNew YorkUSA

Personalised recommendations