Mathematical Foundations of Robust and Distributionally Robust Optimization

Robust and distributionally robust optimization are modeling paradigms for decision-making under uncertainty where the uncertain parameters are only known to reside in an uncertainty set or are governed by any probability distribution from within an ambiguity set, respectively, and a decision is sought that minimizes a cost function under the most adverse outcome of the uncertainty. In this paper, we develop a rigorous and general theory of robust and distributionally robust nonlinear optimization using the language of convex analysis. Our framework is based on a generalized `primal-worst-equals-dual-best' principle that establishes strong duality between a semi-infinite primal worst and a non-convex dual best formulation, both of which admit finite convex reformulations. This principle offers an alternative formulation for robust optimization problems that may be computationally advantageous, and it obviates the need to mobilize the machinery of abstract semi-infinite duality theory to prove strong duality in distributionally robust optimization. We illustrate the modeling power of our approach through convex reformulations for distributionally robust optimization problems whose ambiguity sets are defined through general optimal transport distances, which generalize earlier results for Wasserstein ambiguity sets.

Distributionally Robust Convex Optimization

Daniel Kuhn, Wolfram Wiesemann

Distributionally robust optimization is a paradigm for decision-making under uncertainty where the uncertain problem data is governed by a probability distribution that is itself subject to uncertainty. The distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker's prior information. In this paper, we propose a unifying framework for modeling and solving distributionally robust optimization problems. We introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold. These ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases. They also allow us to characterize distributional families in terms of several classical and/or robust statistical indicators that have not yet been studied in the context of robust optimization. We determine sharp conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable. We also provide tractable conservative approximations for problems that violate these conditions.

2014

Decision rules for information discovery in multi-stage stochastic programming

Daniel Kuhn

Stochastic programming and robust optimization are disciplines concerned with optimal decision-making under uncertainty over time. Traditional models and solution algorithms have been tailored to problems where the order in which the uncertainties unfold is independent of the controller actions. Nevertheless, in numerous real-world decision problems, the time of information discovery can be influenced by the decision maker, and uncertainties only become observable following an (often costly) investment. Such problems can be formulated as mixed-binary multi-stage stochastic programs with decision-dependent non-anticipativity constraints. Unfortunately, these problems are severely computationally intractable. We propose an approximation scheme for multi-stage problems with decision-dependent information discovery which is based on techniques commonly used in modern robust optimization. In particular, we obtain a conservative approximation in the form of a mixed-binary linear program by restricting the spaces of measurable binary and real-valued decision rules to those that are representable as piecewise constant and linear functions of the uncertain parameters, respectively. We assess our approach on a problem of infrastructure and production planning in offshore oil fields from the literature.

IEEE2011

From Data to Decisions: Distributionally Robust Optimization is Optimal

Daniel Kuhn, Peyman Mohajerin Esfahani

We study stochastic programs where the decision-maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, i.e., a predictor, and an optimizer of the estimated cost function that serves as a near-optimal candidate decision, i.e., a prescriptor. As functions of the data, predictors and prescriptors constitute statistical estimators. We propose a meta-optimization problem to find the least conservative predictors and prescriptors subject to constraints on their out-of-sample disappointment. The out-of-sample disappointment quantifies the probability that the actual expected cost of the candidate decision under the unknown true distribution exceeds its predicted cost. Leveraging tools from large deviations theory, we prove that this meta-optimization problem admits a unique solution: The best predictor-prescriptor pair is obtained by solving a distributionally robust optimization problem over all distributions within a given relative entropy distance from the empirical distribution of the data.

2020