Likelihood robust optimization for data-driven problems
- 1.6k Downloads
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.
KeywordsEmpirical Distribution Robust Optimization Convex Optimization Problem Bayesian Statistic Daily Return
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.
- Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton Series in Applied Mathematics. Princeton University Press, PrincetonGoogle Scholar
- Bertsimas D, Thiele A (2006) Robust and data-driven optimization: modern decision-making under uncertainty. In: Tutorial on Operations Research, INFORMSGoogle Scholar
- Bertsimas D, Gupta V, Kallus N (2013) Data-driven robust optimization. arXiv:1401.0212
- Delage E (2009) Distributionally robust optimization in context of data-driven problems. PhD thesisGoogle Scholar
- Gelman A, Carlin J, Stern HS, Rubin DB (1995) Bayesian data analysis. Chapman and Hall Press, New YorkGoogle Scholar
- Johnson N, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol. 1. Wiley Series in Probability and StatisticsGoogle Scholar
- Luenberger D (1997) Investment science. Oxford University Press, OxfordGoogle Scholar
- 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
- Shapiro A, Dentcheva D, Ruszczynski A (2009) Lectures on stochastic programming: modeling and theory. MPS-SIAM Series on OptimizationGoogle Scholar
- Wassaman L (2009) All of statistics: a concise course in statistical inference. Springer Texts in Statistics, New YorkGoogle Scholar
- Ẑáĉková J (1966) On minimax solutions of stochastic linear programming problems. Casopis pro Pêstování Matematiky 91:423–430Google Scholar