Data-driven Distributionally Robust Optimization Using the Wasserstein\n Metric: Performance Guarantees and Tractable Reformulations

2015/05/19 by Peyman Mohajerin Esfahani, Daniel Kühn, Esfahani, Peyman Mohajerin +1 · 67 citations
Decision Sciences · Economics, Econometrics and Finance · #Risk and Portfolio Optimization #Market Dynamics and Volatility

paper · pdf · doi:10.48550/arxiv.1505.05116

Abstract

We consider stochastic programs where the distribution of the uncertain\nparameters is only observable through a finite training dataset. Using the\nWasserstein metric, we construct a ball in the space of (multivariate and\nnon-discrete) probability distributions centered at the uniform distribution on\nthe training samples, and we seek decisions that perform best in view of the\nworst-case distribution within this Wasserstein ball. The state-of-the-art\nmethods for solving the resulting distributionally robust optimization problems\nrely on global optimization techniques, which quickly become computationally\nexcruciating. In this paper we demonstrate that, under mild assumptions, the\ndistributionally robust optimization problems over Wasserstein balls can in\nfact be reformulated as finite convex programs---in many interesting cases even\nas tractable linear programs. Leveraging recent measure concentration results,\nwe also show that their solutions enjoy powerful finite-sample performance\nguarantees. Our theoretical results are exemplified in mean-risk portfolio\noptimization as well as uncertainty quantification.\n

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