Computational Management Science

, Volume 13, Issue 2, pp 241–261 | Cite as

Likelihood robust optimization for data-driven problems

  • Zizhuo WangEmail author
  • Peter W. Glynn
  • Yinyu Ye
Original Paper


We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.


Empirical Distribution Robust Optimization Convex Optimization Problem Bayesian Statistic Daily Return 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank Dongdong Ge and Zhisu Zhu for valuable insights and discussions. The authors also thank Erick Delage for insightful discussions and for sharing useful codes for the numerical experiments. The research of the first author is supported by the National Science Foundation (NSF) under research Grant CMMI-1434541.


  1. Ben-Tal A, den Hertog D, De Waegenaere A, Melenberg B, Rennen G (2013) Robust solutions of optimization problems affected by uncertain probabilities. Manage Sci 59(2):341–357CrossRefGoogle Scholar
  2. Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton Series in Applied Mathematics. Princeton University Press, PrincetonGoogle Scholar
  3. Ben-Tal A, Nemirovski A (1999) Robust solutions of uncertain linear programs. Oper Res Lett 25(1):1–13CrossRefGoogle Scholar
  4. Bertsimas D, Thiele A (2006) Robust and data-driven optimization: modern decision-making under uncertainty. In: Tutorial on Operations Research, INFORMSGoogle Scholar
  5. Bertsimas D, Brown D, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501CrossRefGoogle Scholar
  6. Bertsimas D, Gupta V, Kallus N (2013) Data-driven robust optimization. arXiv:1401.0212
  7. Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53CrossRefGoogle Scholar
  8. Bertsimas D, Thiele A (2006) A robust optimization approach to inventory theory. Oper Res 54(1):150–168CrossRefGoogle Scholar
  9. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. Calafiore G, El Ghaoui L (2006) On distributionally robust chance-constrained linear programs. J Optim Theory Appl 130(1):1–22CrossRefGoogle Scholar
  11. Chen X, Sim M, Sun P (2007) A robust optimization perspective on stochastic programming. Oper Res 55(6):1058–1071CrossRefGoogle Scholar
  12. Delage E (2009) Distributionally robust optimization in context of data-driven problems. PhD thesisGoogle Scholar
  13. Delage E, Ye Y (2008) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper Res 58(3):595–612CrossRefGoogle Scholar
  14. Dupaĉová J (1987) The minimax approach to stochastic programming and an illustrative application. Stochastics 20(1):73–88CrossRefGoogle Scholar
  15. Gabrel V, Murat C, Thiele A (2014) Recent advances in robust optimization: An overview. Eur J Oper Res 235(3):471–483CrossRefGoogle Scholar
  16. Gallego G, Moon I (1993) The distribution free newsboy problem: Review and extension. J Oper Res Soc 44(8):825–834CrossRefGoogle Scholar
  17. Gelman A, Carlin J, Stern HS, Rubin DB (1995) Bayesian data analysis. Chapman and Hall Press, New YorkGoogle Scholar
  18. Iyengar G (2005) Robust dynamic programming. Math Oper Res 30(2):257–280CrossRefGoogle Scholar
  19. Johnson N, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol. 1. Wiley Series in Probability and StatisticsGoogle Scholar
  20. Khouja M (1999) The single period newsvendor problem: literature review and suggestions for future research. Omega 27:537–553CrossRefGoogle Scholar
  21. Luenberger D (1997) Investment science. Oxford University Press, OxfordGoogle Scholar
  22. Mason D, Schuenemeyer J (1983) A modified \(\text{ K }\)olmogorov-\(\text{ S }\)mirnov test sensitive to tail alternatives. Ann Stat 11(3):933–946CrossRefGoogle Scholar
  23. Nilim A, El Ghaoui L (2005) Robust control of Markov decision processes with uncertain transition matrices. Oper Res 53(5):780–798CrossRefGoogle Scholar
  24. Owen A (2001) Empirical likelihood. Chapman and Hall Press, LondonCrossRefGoogle Scholar
  25. Pardo L (2005) Statistical inference based on divergence measures. Chapman and Hall Press, LondonCrossRefGoogle Scholar
  26. Perakis G, Roels G (2008) Regret in the newsvendor model with partial information. Oper Res 56(1):188–203CrossRefGoogle Scholar
  27. Popescu I (2007) Robust mean-covariance solutions for stochastic optimization. Oper Res 55(1):98–112CrossRefGoogle Scholar
  28. Scarf H (1958) A min-max solution of an inventory problem. In: Arrow K, Karlin S, Scarf H (eds) Studies in the mathematical theory of inventory and production. Stanford University Press, California, pp 201–209Google Scholar
  29. Scarf H (1959) Bayes solutions of the statistical inventory problem. Ann Math Stat 30(2):490–508CrossRefGoogle Scholar
  30. Shapiro A, Dentcheva D, Ruszczynski A (2009) Lectures on stochastic programming: modeling and theory. MPS-SIAM Series on OptimizationGoogle Scholar
  31. Shapiro A, Kleywegt AJ (2002) Minimax analysis of stochastic programs. Optim Methods Softw 17:523–542CrossRefGoogle Scholar
  32. So AMS, Zhang J, Ye Y (2009) Stochastic combinatorial optimization with controllable risk aversion level. Math Oper Res 34(3):522–537CrossRefGoogle Scholar
  33. Wassaman L (2009) All of statistics: a concise course in statistical inference. Springer Texts in Statistics, New YorkGoogle Scholar
  34. Yue J, Chen B, Wang M (2006) Expected value of distribution information for the newsvendor problem. Oper Res 54(6):1128–1136CrossRefGoogle Scholar
  35. Ẑáĉková J (1966) On minimax solutions of stochastic linear programming problems. Casopis pro Pêstování Matematiky 91:423–430Google Scholar
  36. Zhu Z, Zhang J, Ye Y (2013) Newsvendor optimization with limited distribution information. Optim Methods Softw 28(3):640–667CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Management Science and EngineeringStanford UniversityStanfordUSA

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