[Paper Review] Distributionally Robust Optimization: A Review

Hamed Rahimian, Sanjay Mehrotra|arXiv (Cornell University)|Aug 13, 2019

Risk and Portfolio Optimization269 references332 citations

TL;DR

A comprehensive survey of distributionally robust optimization (DRO), outlining its formulations, connections to related concepts, solution methods, ambiguity-set models, calibration, and applications in operations research and machine learning.

ABSTRACT

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization.

Motivation & Objective

• Motivation: uncertainty in underlying probability distributions and the need to hedge against distributional ambiguity.
• Goal: provide a holistic review of DRO, linking it with robust optimization, risk-averse optimization, chance constraints, and regularization.
• Aim: categorize ambiguity-set models, discuss solution methods, and cover calibration and applications in OR and ML.

Proposed method

• Present a general DRO model and show how it unifies SO and RO.
• Discuss solution techniques including cutting-surface methods and dual methods.
• Survey ambiguity-set constructions: discrepancy-based, moment-based, shape-preserving, kernel-based, and general sets.
• Explain calibration of robustness parameters and data-driven DRO approaches.
• Relate DRO to game theory, risk measures, and regularization to elucidate connections.
• Define and distinguish objective and constraint handling via worst-case expectations over P in the ambiguity set.

Experimental results

Research questions

• RQ1 How can DRO be formulated to hedge against distributional ambiguity while interpolating between SO and RO?
• RQ2 What are the main families of ambiguity sets, and how do they impact tractability and robustness?
• RQ3 How are DRO models solved algorithmically, and what are the roles of cutting-plane/duality methods?
• RQ4 How should ambiguity-set parameters be calibrated, including data-driven approaches?
• RQ5 What are the relationships between DRO and related concepts like risk measures, chance constraints, and regularization?

Key findings

• DRO provides a unifying framework between stochastic and robust optimization by optimizing over a family of distributions.
• Two primary solution approaches are cutting-surface methods and dual methods for semi-infinite or robust reformulations.
• Ambiguity sets can be constructed via discrepancies, moments, shapes, kernels, or general specifications, enabling flexible modeling of distributional uncertainty.
• Calibration of nominal parameters and robustness levels can be data-driven or non-data-driven, influencing conservatism and performance.
• DRO links to risk-averse optimization and coherent/law-invariant risk measures, and relates to regularization in statistical learning.
• The framework accommodates continuous and finite sample spaces and supports out-of-sample performance guarantees and asymptotic consistency.