The main focus of this paper is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints. Our eventual goal is to handle supplier problems in the most general distributional setting, where there is only black-box access to the underlying distribution. To that end, we follow a two-step approach. First, we develop algorithms for a restricted version of each problem, where all scenarios are explicitly provided; second, we employ a novel scenario-discarding variant of the standard Sample Average Approximation (SAA) method, which crucially exploits properties of the restricted-case algorithms. We note that the scenario-discarding modification to the SAA method is necessary in order to optimize over the radius.

翻译：本文主要研究两阶段随机设置下的半径基（供应商）聚类问题，其中模型的固有随机性表现为预算约束。我们还探讨了若干变体问题，这些变体对第一阶段决策施加了额外约束，具体包括拟阵约束和多背包约束。我们的最终目标是在最一般的分布设置下处理供应商问题，即仅能通过黑箱方式访问底层分布。为此，我们采用两步法：首先，针对每个问题的受限版本（其中所有情景均明确给出）开发算法；其次，采用标准样本均值近似方法的创新性情景丢弃变体，该变体关键性地利用了受限案例算法的性质。我们指出，为优化半径，必须对SAA方法进行情景丢弃的修改。