Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems

  • Authors:

  • Erick Delage;Yinyu Ye

  • Affiliations:

  • Department of Management Sciences, HEC Montréal, Montreal, Quebec H3T 2A7, Canada;Department of Management Science and Engineering, Stanford University, Stanford, California 94305

  • Venue:
  • Operations Research
  • Year:
  • 2010

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Abstract

Stochastic programming can effectively describe many decision-making problems in uncertain environments. Unfortunately, such programs are often computationally demanding to solve. In addition, their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix). We demonstrate that for a wide range of cost functions the associated distributionally robust (or min-max) stochastic program can be solved efficiently. Furthermore, by deriving a new confidence region for the mean and the covariance matrix of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. These arguments are confirmed in a practical example of portfolio selection, where our framework leads to better-performing policies on the “true” distribution underlying the daily returns of financial assets.