AI 中文总结
该研究提出将预测融入鲁棒调度的一致性-鲁棒性框架，针对不同不确定性模型与机器环境得到相应权衡结果，明确结合一致性与鲁棒性的可能性取决于二者的相互作用。
AI 中文摘要
鲁棒优化通过在规定的不确定性集合上针对最坏情况进行优化来应对不确定性，但若预测、历史数据或学习到的预测表明存在更可能的场景，这种保护可能会过于保守。我们提出一种带预测的鲁棒优化框架，输入包括一个不确定性集合和一个特定的预测场景，目标是计算出一个既具有一致性（即对预测场景接近最优）又具有鲁棒性（即与经典极小极大鲁棒最优解具有竞争力）的单一解。与标准学习增强算法不同，预测不仅是对已实现输入的估计，还会生成一个独立的基准——预测最优解，该基准需与极小极大鲁棒最优解进行权衡。我们针对处理时间不确定的完工时间调度问题研究此框架，并对标准不确定性模型和机器环境进行结构分类：对于区间不确定性，我们针对受限分配及相关机器得到了平滑的$(1+1/\boldsymbol{\u03BB},1+\boldsymbol{\u03BB})$一致性-鲁棒性权衡；此外，我们证明无关机器不存在常数权衡。对于预算不确定性，我们针对受限分配得到了$(1+1/\boldsymbol{\u03BB},2+\boldsymbol{\u03BB})$权衡。我们的分析基于对偶性归约至类区间上包络，辅以一个下界证明：即使仅有一个作业可能偏离，相关机器也不存在常数权衡。对于任意不确定性集合，我们通过支撑函数块构造为相同机器得到了常数权衡，并证明受限分配情况下不存在可行性。我们的结果表明，鲁棒调度中结合一致性与鲁棒性的可能性关键取决于不确定性模型与机器环境之间的相互作用。
英文摘要
Robust optimization protects against uncertainty by optimizing for the worst case over a prescribed uncertainty set. This protection can be overly conservative when forecasts, historical data, or learned predictions indicate a more likely scenario. We introduce a framework for robust optimization with predictions. The input consists of an uncertainty set together with a distinguished predicted scenario, and the goal is to compute a single solution that is both consistent, meaning near-optimal for the predicted scenario, and robust, meaning competitive with the classical min-max robust optimum. Unlike in standard learning-augmented algorithms, the prediction does not merely estimate the realized input; it creates a separate benchmark, the predicted optimum, which must be balanced against the min-max robust optimum. We study this framework for makespan scheduling with uncertain processing times and give a structural classification across standard uncertainty models and machine environments. For interval uncertainty, we obtain a smooth $(1+1/λ,1+λ)$ consistency-robustness tradeoff for restricted-assignment and related machines. Furthermore, we prove that unrelated machines admit no constant tradeoff. For budgeted uncertainty, we obtain a $(1+1/λ,2+λ)$ tradeoff for restricted assignment. Our analysis is based on a duality-based reduction to an interval-like upper envelope. We complement this with a lower bound showing that related machines admit no constant tradeoff even when only one job may deviate. For arbitrary uncertainty sets, we obtain constant tradeoffs for identical machines via a support-function block construction, and prove impossibility for restricted assignment. Our results show that the possibility of combining consistency and robustness in robust scheduling depends critically on the interaction between the uncertainty model and the machine environment.
