Optimized Dimensionality Reduction for Moment-Based Distributionally Robust Optimization

Most moment-based distributionally robust optimization (DRO) problems can be reformulated as semidefinite programming (SDP) problems, which can be solved in polynomial time. However, solving high-dimensional SDPs is often time-consuming. Existing approximation methods typically reduce the dimensionality of random parameters before solving the approximated SDPs. This sequential approach relies solely on statistical information to reduce the high-dimensional uncertainty space, which may not yield the best approximation performance. Jiang et al. (2025) introduce an optimized dimensionality reduction (ODR) approach that integrates the dimensionality reduction of random parameters with subsequent optimization problems. This integration enables two outer approximations and one inner approximation of the original problem, all represented as low-dimensional SDPs that can be solved efficiently, providing two lower bounds and one upper bound, respectively. More importantly, these approximations can theoretically achieve the optimal value of the original high-dimensional SDPs, resulting in a zero gap.