Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

Abstract

In optimization problems appearing in fields such as economics, finance, or engineering, it is often important that a risk measure of a decision-dependent random variable stays below a prescribed level. At the same time, the underlying probability distribution determining the risk measure's value is typically known only up to a certain degree and the constraint should hold for a reasonably wide class of probability distributions. In addition to that, the constraint should be computationally tractable. In this paper we review and generalize results on the derivation of tractable counterparts of such constraints for discrete probability distributions. Using established techniques in robust optimization, 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. This holds for a wide range of risk measures and uncertainty sets for probability distributions defined using statistical goodness-of-fit tests or probability metrics. In this way, we provide a unified framework of reformulating this class of constraints, extending the number of solvable risk measure-uncertainty set combinations considerably, including also risk measures that are nonlinear in the probabilities. To provide a clear overview for the user, we give the computational tractability status for each of the uncertainty set-risk measure pairs of which some have been solved in the literature. Examples, including portfolio optimization and antenna array design, illustrate the proposed approach in a theoretical and numerical setting. block includes the following risk measures: negative mean return, Optimized Certainty Equivalent (with Conditional Value-at-Risk as a special case), Certainty Equivalent, Shortfall Risk, lower partial moments, mean absolute deviation from the median, standard deviation/variance less mean, Sharpe ratio, and the Entropic Value-at-Risk. The second building block encompasses uncertainty sets built using φ-divergences (with the Pearson (χ 2 ) and likelihood ratio (G) tests as special cases), Kolmogorov-Smirnov test, Wasserstein (Kantorovich) distance, Anderson-Darling, Cramer-von Mises, Watson, and Kuiper tests.Secondly, we address a common feature rendering many optimization constraints SOCP SOCP QP Certainty Equivalent Shortfall risk • CP [10],[11] • CP [10][11][38] • CP [10][11][34][54] CP CP • CP [10] CP CP CP Optimized Certainty Equivalent CP CP CP CP CP CP CP CP CP Value-at-Risk • [35] • [35] • [34][35][37] Conditional Value-at-Risk CP • SOCP [14][55] • CP [14][34] • LP [14][55] • LP [45][55][56] • CP [14] • SOCP [14][55] • SOCP [14][55] • LP [14][55] Entropic Value-at-Risk CP CP CP CP CP CP CP CP CP Mean absolute deviation from the mean • [17] Distortion risk measures • [45][56] Coherent risk measures Spectral risk measures

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