Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

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Abstract

In this paper we study distributionally robust constraints on risk measures (such
as standard deviation less the mean, Conditional Value-at-Risk, Entropic Value-at-Risk) of decision-dependent random variables. The uncertainty sets for the discrete probability distributions are defined using statistical goodness-of-fit tests and probability metrics such as Pearson, likelihood ratio, Anderson-Darling tests, or Wasserstein distance. This type of constraints arises in problems in portfolio optimization, economics, machine learning, and engineering. We show that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set. We also show how the counterpart can be constructed for risk measures that are nonlinear in the probabilities (for example, variance or the Conditional Value-at-Risk). We provide the computational tractability status for each of the uncertainty set-risk measure pairs that we could solve. Numerical examples including portfolio optimization and a multi-item newsvendor problem illustrate the proposed approach.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages39
Volume2014-031
Publication statusPublished - 9 May 2014

Publication series

NameCentER Discussion Paper
Volume2014-031

Fingerprint

Risk Measures
Conditional Value at Risk
Portfolio Optimization
Uncertainty
Anderson-Darling Test
Newsvendor Problem
Probability Metrics
Wasserstein Distance
Dependent Random Variables
Value at Risk
Tractability
Discrete Distributions
Goodness of Fit Test
Likelihood Ratio
Statistical test
Standard deviation
Machine Learning
Probability Distribution
Economics
Engineering

Keywords

  • risk measure
  • robust counterpart
  • nonlinear inequality
  • robust optimization
  • support functions

Cite this

Postek, K. S., den Hertog, D., & Melenberg, B. (2014). Tractable Counterparts of Distributionally Robust Constraints on Risk Measures. (CentER Discussion Paper; Vol. 2014-031). Tilburg: Operations research.
Postek, K.S. ; den Hertog, D. ; Melenberg, B. / Tractable Counterparts of Distributionally Robust Constraints on Risk Measures. Tilburg : Operations research, 2014. (CentER Discussion Paper).
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Postek, KS, den Hertog, D & Melenberg, B 2014 'Tractable Counterparts of Distributionally Robust Constraints on Risk Measures' CentER Discussion Paper, vol. 2014-031, Operations research, Tilburg.

Tractable Counterparts of Distributionally Robust Constraints on Risk Measures. / Postek, K.S.; den Hertog, D.; Melenberg, B.

Tilburg : Operations research, 2014. (CentER Discussion Paper; Vol. 2014-031).

Research output: Working paperDiscussion paperOther research output

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AB - In this paper we study distributionally robust constraints on risk measures (suchas standard deviation less the mean, Conditional Value-at-Risk, Entropic Value-at-Risk) of decision-dependent random variables. The uncertainty sets for the discrete probability distributions are defined using statistical goodness-of-fit tests and probability metrics such as Pearson, likelihood ratio, Anderson-Darling tests, or Wasserstein distance. This type of constraints arises in problems in portfolio optimization, economics, machine learning, and engineering. We show that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set. We also show how the counterpart can be constructed for risk measures that are nonlinear in the probabilities (for example, variance or the Conditional Value-at-Risk). We provide the computational tractability status for each of the uncertainty set-risk measure pairs that we could solve. Numerical examples including portfolio optimization and a multi-item newsvendor problem illustrate the proposed approach.

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Postek KS, den Hertog D, Melenberg B. Tractable Counterparts of Distributionally Robust Constraints on Risk Measures. Tilburg: Operations research. 2014 May 9. (CentER Discussion Paper).