机器学习和人工智能中的数学规划：模型与应用的统一分类法
Mathematical Programming in Machine Learning and Artificial Intelligence: A Unified Taxonomy of Models and Applications
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中文总结 AI 辅助
本文提出一种统一分类法，将机器学习与AI中的各类应用按数学规划范式（如线性、混合整数、双级等）组织，并统一符号、比较可处理性与求解策略，强调其作为预测与约束决策间纪律性接口的价值。
中文摘要 AI 辅助
数学规划为现代机器学习（ML）和人工智能（AI）系统中嵌入的许多决策提供了一种通用语言：选择检索上下文、路由令牌、分配推理计算、拟合结构化预测器、防止分布偏移以及平衡相互竞争的目标。然而，相关文献分散在优化、信息检索、推荐、自然语言处理、计算机视觉和学习理论等领域。本文在各种常见的数学规划范式下组织了各种应用：线性、二次、二元和混合整数、锥、双级、多目标、逆、分布鲁棒、子模和极小极大优化。我们使用基本统一的符号对模型进行规范化，并为每个应用确定输入、决策变量、主要公式、结构属性、求解策略和局限性。跨范式，我们比较了可处理性、松弛质量、分解、近似保证和可扩展性瓶颈。本文表明，数学规划最有用的地方不在于声称所有学习都是LP或MIP，而在于作为预测和约束决策之间的一个纪律性接口。
英文摘要
Mathematical programming provides a common language for many decisions embedded in modern machine-learning (ML) and artificial-intelligence (AI) systems: selecting retrieval context, routing tokens, allocating inference compute, fitting structured predictors, protecting against distribution shift, and balancing competing objectives. However, the relevant literature is fragmented across optimization, information retrieval, recommendation, natural-language processing, computer vision, and learning theory. This paper organizes various applications under common mathematical programming paradigms: linear, quadratic, binary and mixed-integer, conic, bilevel, multi-objective, inverse, distributionally robust, submodular, and min--max optimization. We normalize the models with a mostly unified notation and, for every application, identify inputs, decision variables, a principal formulation, structural properties, solution strategies, and limitations. Across paradigms, we compare tractability, relaxation quality, decomposition, approximation guarantees, and scalability bottlenecks. The paper shows that mathematical programming is most useful not as a claim that all learning is LP or MIP, but as a disciplined interface between predictions and constrained decisions.
