A Soft Robust Model for Optimization Under Ambiguity

  • Authors:

  • Aharon Ben-Tal;Dimitris Bertsimas;David B. Brown

  • Affiliations:

  • William Davidson Faculty of Industrial Engineering and Management, Technion---Israel Institute of Technology, Technion City, Haifa 32000, Israel;Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139;The Fuqua School of Business, Duke University, Durham, North Carolina 27708

  • Venue:
  • Operations Research
  • Year:
  • 2010

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Abstract

In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate the methodology on an asset allocation example consisting of historical market data over a 25-year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk-return trade-off: each case results in a higher out-of-sample expected return for a relatively minor degradation of out-of-sample downside performance.