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Efficient Algorithms for Distributionally Robust Stochastic Optimization with Discrete Scenario Support
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2021-07-13 , DOI: 10.1137/19m1290115
Zhe Zhang , Shabbir Ahmed , Guanghui Lan

SIAM Journal on Optimization, Volume 31, Issue 3, Page 1690-1721, January 2021.
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.


中文翻译:

具有离散场景支持的分布式鲁棒随机优化的高效算法

SIAM 优化杂志,第 31 卷,第 3 期,第 1690-1721 页,2021 年 1 月。
最近,人们对分布式鲁棒优化 (DRO) 作为数据驱动决策制定的原则方法越来越感兴趣。在本文中,我们考虑了具有离散场景支持的分布式鲁棒两阶段随机优化问题。虽然大量的研究工作致力于 DRO 问题的易于处理的重新表述,特别是那些具有连续场景支持的问题,但很少开发出有效的数值算法,并且它们中的大多数既不能处理非光滑的第二阶段成本函数,也不能处理大量的场景 $K $ 有效。我们通过将 DRO 问题重新表述为三线性 min-max-max 鞍点问题并开发新算法来填补这一空白,该算法可以实现 $\mathcal{O}(1/\epsilon)$ 迭代复杂度,该复杂度仅轻微依赖于 $K美元。如有必要,这些算法的每次迭代中涉及的主要计算可以并行进行。此外,为了使用 Kantorovich 球模糊集解决一类重要的 DRO 问题,我们建议对我们的算法进行轻微修改,以避免以 $\mathcal{O}(\sqrt{ K})$ 次迭代。最后,进行了初步的数值实验,以证明所提出算法的经验优势。我们建议对我们的算法进行轻微修改,以避免以 $\mathcal{O}(\sqrt{K})$ 次迭代为代价进行昂贵的概率向量投影计算。最后,进行了初步的数值实验,以证明所提出算法的经验优势。我们建议对我们的算法进行轻微修改,以避免以 $\mathcal{O}(\sqrt{K})$ 次迭代为代价进行昂贵的概率向量投影计算。最后,进行了初步的数值实验,以证明所提出算法的经验优势。
更新日期:2021-07-13
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