Tractable Distributionally Robust Optimization with Data (original) (raw)
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INFORMS Journal on Computing
Distributionally robust optimization (DRO) is a modeling framework in decision making under uncertainty in which the probability distribution of a random parameter is unknown although its partial information (e.g., statistical properties) is available. In this framework, the unknown probability distribution is assumed to lie in an ambiguity set consisting of all distributions that are compatible with the available partial information. Although DRO bridges the gap between stochastic programming and robust optimization, one of its limitations is that its models for large-scale problems can be significantly difficult to solve, especially when the uncertainty is of high dimension. In this paper, we propose computationally efficient inner and outer approximations for DRO problems under a piecewise linear objective function and with a moment-based ambiguity set and a combined ambiguity set including Wasserstein distance and moment information. In these approximations, we split a random ve...
Distributionally Robust Optimization with Decision-Dependent Ambiguity Set
2018
Abstract: We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover’s distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest, and provide an overview of various settings leading to tractable formulations. Some of the arising side results are also of independent interest, including mathematical programming expressions for robustified risk measures in a discrete space. Finally, we rely on state-of-the-art modeling techniques from machine scheduling and humanitarian logistics to arrive at potentially practical applications.
Computationally Efficient Approximations for Distributionally Robust Optimization
Optimization Online, 2020
Distributionally robust optimization (DRO) is a modeling framework in decision making under uncertainty where the probability distribution of a random parameter is unknown while its partial information (e.g., statistical properties) is available. In this framework, the unknown probability distribution is assumed to lie in an ambiguity set consisting of all distributions that are compatible with the available partial information. Although DRO bridges the gap between stochastic programming and robust optimization, one of its limitations is that its models for large-scale problems can be siginificantly difficult to solve, especially when the uncertainty is of high dimension. In this paper, we propose computationally efficient inner and outer approximations for DRO problems with a moment-based ambiguity set and a combined ambiguity set including Wasserstein distance and moment information. In these approximations, we split a random vector into smaller pieces, leading to smaller matrix constraints. In addition, we use principal component analysis to shrink uncertainty space dimensionality. We quantify the quality of the developed approximations by deriving theoretical bounds on their optimality gap. We display the practical applicability of the proposed approximations in a production-transportation problem and a multi-product newsvendor problem. The results demonstrate that these approximations dramatically reduce the computational time while maintaining high solution quality.
Distributionally Robust Optimization with Expected Constraints via Optimal Transport
2021
We consider a stochastic program with expected value constraints. We analyze this problem in a general context via Distributionally Robust Optimization (DRO) approach using 1 or 2-Wasserstein metrics where the ambiguity set depends on the decision. We show that this approach can be reformulated as a finite-dimensional optimization problem, and, in some cases, this can be convex. Additionally, we establish criteria to determine the feasibility of the problem in terms of the Wasserstein radius and the level of the constraint. Finally, we present numerical results in the context of management inventory and portfolio optimization. In portfolio optimization context, we present the advantages that our approach has over some existing non-robust methods using real financial market data.
2021
We study two-stage stochastic optimization problems with random recourse, where the adaptive decisions are multiplied with the uncertain parameters in both the objective function and the constraints. To mitigate the computational intractability of infinite-dimensional optimization, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We then develop a data-driven distributionally robust framework with two layers of robustness to address distributionally uncertainty. The emerging optimization problem can be reformulated as an exact copositive program, which admits tractable approximations in semidefinite programming. We design a decomposition algorithm where smaller-size semidefinite programs can be solved in parallel, which further reduces the runtime. Lastly, we establish the performance guarantees of the proposed scheme and demonstrate its effectiveness through numerical examples.
2020
Abstract: We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets, we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover’s distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest, and provide an overview of various settings leading to tractable formulations. Some of the arising side results, such as the mathematical programming expressions for robustified risk measures in a discrete space, are also of independent interest. Finally, we rely on state-of-the-art modeling techniques from humanitarian logistics and machine scheduling to arrive at potentially practical applications, and present a numerical study for a novel risk-averse scheduling problem with controllable processing times.