AI 中文总结
本文针对Γ-鲁棒多选背包问题，提出 simultaneous 组-包络界方法，可同时界定阈值族，大幅加速松弛求解，在实验中实现2.37倍几何平均加速。
AI 中文摘要
许多鲁棒规划问题通过针对每个不确定性阈值求解一族普通优化问题来解决，反复构建和求解这些松弛问题会主导运行时间。本文研究当决策从每个组中恰好选择一个选项时，多选背包松弛问题的整个阈值族可被同时界定：通过消去移除阈值特定基线，将每个组的贡献简化为两个简单包络；排序后，在与阈值数量线性、选项数量近线性的时间内对所有阈值评估一个乘子，得到的区间界有效，与单个可行阈值下的普通松弛匹配，且至少与匹配的组-团比较器一样强。本文采用经验证的一维搜索控制乘子误差，通过精确检查解决数值模糊的可行性与比较情况。在相同自适应搜索中，该方法在全部60个实例级中位数比较中获胜，实现2.37倍几何平均加速；额外消融实验、独立线性规划检查及从公开档案转换的9个实例（超出内部生成器范围）验证了该机制，单独整数审计显示：当整数搜索占主导时，更快的松弛界未必有用。
英文摘要
Many robust planning problems are solved by checking a family of ordinary optimization problems, one for each uncertainty threshold. Repeatedly building and solving those relaxations can dominate runtime. We show that, when a decision chooses exactly one option from each group, the entire threshold family of multiple-choice knapsack relaxations can instead be bounded together. A cancellation removes threshold-specific baselines and reduces each group's contribution to two simple envelopes. After sorting, one multiplier is evaluated across all thresholds in time linear in their number and nearly linear in the number of options. The resulting interval bound is valid, matches the ordinary relaxation on a single feasible threshold, and is at least as strong as the matched group-clique comparator. A certified one-dimensional search controls multiplier error, while exact checks resolve numerically ambiguous feasibility and comparison cases. In an otherwise identical adaptive search, the method wins all 60 instance-level median comparisons and achieves a 2.37-fold geometric-mean speedup. Additional ablations, independent linear-program checks, and nine instances transformed from a published archive test the mechanism beyond the internal generator. A separate integer audit shows the boundary: faster relaxation bounds need not help when integer search dominates.
Comments19 pages, 2 figures. Code and reproducibility materials: https://github.com/eric939/simultaneous-group-envelope-mckp
