发表机构
Northwestern University（西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究分布鲁棒线性机会约束问题，用高斯混合模型建模不确定性。提出基于连续参数空间Wasserstein-2模糊集的方法，证明强对偶性并推导半无限重新表述，开发算法，通过案例研究展示其在实现可靠性目标及引起能量分配结构变化方面的价值。
AI 中文摘要
我们研究分布鲁棒线性机会约束问题，其中不确定性由高斯混合模型（GMM）建模。有限支持分布鲁棒（FDR）公式在数据驱动的鲁棒优化中广泛使用，它在经验混合支持点上进行鲁棒化，主要是对拟合的名义混合进行压力测试。当服务可靠性取决于名义混合支持参数的结构错误指定时，这可能不够。为了解决这一限制，我们通过开发一种新颖的Wasserstein-2度量公式来描述分布的模糊集，该公式使用具有有限二阶矩的概率测度上的Bures-Wasserstein（BW）度量。与通常先验设置有限多个经验支持点的FDR不同，所提出的模糊集允许最坏情况分布内生地确定有多少混合成分获得质量以及它们的均值和协方差在连续支持内的位置。对于由此产生的模糊集，在温和的正则条件下，我们证明了内部最坏情况机会约束问题的强对偶性，并推导了其半无限重新表述。然后我们开发了一种自适应切割平面算法，该算法内生地确定获得质量的混合成分的位置以及这些位置处高斯分布的均值和协方差。该算法在有限多次迭代中达到任何规定的最优性差距，而块交替局部搜索识别新的成分。使用电动汽车充电站能量分配问题的案例研究证明了该框架在实现任何可靠性目标方面的实用价值。与FDR不同，FDR的分配保持接近名义解，CDR还会引起能量分配的结构变化。
英文摘要
We study distributionally robust linear chance-constrained problems in which uncertainty is modeled by a Gaussian mixture model (GMM). Finite-support distributionally robust (FDR) formulations, widely used in data-driven robust optimization, robustify over empirical mixture support points and therefore primarily stress-test the fitted nominal mixture. This can be insufficient when service reliability depends on structural misspecification of the nominal mixture-support parameters. To address this limitation, we describe the ambiguity set of distributions by developing a novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments. Unlike FDR, which generally sets finitely many empirical support points a priori, the proposed ambiguity set allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support. For the resulting ambiguity set, under mild regularity conditions, we prove strong duality for the inner worst-case chance-constraint problem and derive its semi-infinite reformulation. We then develop an adaptive cutting-surface algorithm, which endogenously determines the locations of mixture components receiving mass, and the mean and covariances of the Gaussian distributions at these locations. The algorithm attains any prescribed optimality gap in finitely many iterations, while a block-alternating local search identifies new components. A case study using the electric-vehicle charging-station energy-allocation problem demonstrates the framework's practical value in achieving any reliability targets. CDR also induces structural changes in energy allocations, unlike FDR, whose allocations remain close to the nominal solution.
