Optimized Dimensionality Reduction for Moment-Based Distributionally Robust Optimization

Shiyi Jiang

doi:10.1287/opre.2023.0645

2025-12-01

PRE

DOI：10.1287/opre.2023.0645

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摘要

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Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most momentbased DRO problems can be reformulated as semidefinite programming (SDP) problems that can be solved in polynomial time, solving high-dimensional SDPs is still timeconsuming. Unlike existing approximation approaches that first reduce the dimensionality of random parameters and then solve the approximated SDPs, we propose an optimized dimensionality reduction (ODR) approach by integrating the dimensionality reduction of random parameters with the subsequent optimization problems. Such integration enables two outer and one inner approximations of the original problem, all of which are lowdimensional SDPs that can be solved efficiently, providing two lower bounds and one upper bound correspondingly. More importantly, these approximations can theoretically achieve the optimal value of the original high-dimensional SDPs. As these approximations are nonconvex SDPs, we develop modified alternating direction method of multipliers algorithms to solve them efficiently. We demonstrate the effectiveness of our proposed ODR approach and algorithm in solving multiproduct newsvendor and production-transportation problems. Numerical results show significant advantages of our approach regarding computational time and solution quality over the three best possible benchmark approaches. Our approach can obtain an optimal or near-optimal (mostly within 0.1%) solution and reduce the computational time by up to three orders of magnitude.

Keyword:

distributionally robust optimization

dimensionality reduction

principal component analysis

semidefinite programming

data-driven optimization