Efficient Algorithms for Distributionally Robust Stochastic Optimization with Discrete Scenario Support
针对离散情景支撑的两阶段分布鲁棒随机优化问题，提出高效算法，迭代复杂度为O(1/ε)且对情景数K依赖小，支持并行计算，并针对Kantorovich球模糊集给出改进版本。
Recently, there has been a growing interest in distributionally robust optimization (DRO) as a principled approach to data-driven decision making. In this paper, we consider a distributionally robust two-stage stochastic optimization problem with discrete scenario support. While much research effort has been devoted to tractable reformulations for DRO problems, especially those with continuous scenario support, few efficient numerical algorithms are developed, and most of them can neither handle the nonsmooth second-stage cost function nor the large number of scenarios $K$ effectively. We fill the gap by reformulating the DRO problem as a trilinear min-max-max saddle point problem and developing novel algorithms that can achieve an $\mathcal{O}(1/\epsilon)$ iteration complexity which only mildly depends on $K$. The major computations involved in each iteration of these algorithms can be conducted in parallel if necessary. Besides, for solving an important class of DRO problems with the Kantorovich ball ambiguity set, we propose a slight modification of our algorithms to avoid the expensive computation of the probability vector projection at the price of an $\mathcal{O}(\sqrt{K})$ times more iterations. Finally, preliminary numerical experiments are conducted to demonstrate the empirical advantages of the proposed algorithms.