Mathematical Programming in Machine Learning and Artificial Intelligence: A Unified Taxonomy of Models and Applications

2026-09-07

AI Summary

本文通过统一的数学规划范式，如线性、二次等优化方法，整理了机器学习和人工智能中决策问题，并比较了各种方法的适用性和局限性。

Abstract

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.

Problem

• Mathematical Programming
• Machine Learning
• Artificial Intelligence
• Optimization
• Applications

Innovation

• mathematical programming
• unified taxonomy
• machine learning
• artificial intelligence
• optimization paradigms