Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

Jose Blanchet (), Karthyek Murthy () and Fan Zhang ()
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Jose Blanchet: Management Science and Engineering, Stanford University, Stanford, California 94305
Karthyek Murthy: Engineering Systems and Design, Singapore University of Technology & Design, Singapore 487372, Singapore
Fan Zhang: Management Science and Engineering, Stanford University, Stanford, California 94305

Mathematics of Operations Research, 2022, vol. 47, issue 2, 1500-1529

Abstract: We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.

Keywords: Primary: 90C15; Secondary: 65K05; 90C47; distributionally robust optimization; stochastic gradient descent; optimal transport; Wasserstein distances; adversarial; strong convexity; comparative statics; rate of convergence (search for similar items in EconPapers)
Date: 2022
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