Mathematical programming organizes decisions in artificial intelligence systems
Mathematical Programming in Machine Learning and Artificial Intelligence: A Unified Taxonomy of Models and Applications
Artificial Intelligence
Summary
Modern AI and machine learning systems make many complex decisions, like where to focus attention or how to use computing power efficiently. The authors explain how different mathematical programming methods can describe these decisions in a unified way. They show how these approaches come from fields like optimization and computer vision and help understand the challenges and strategies for solving problems. This work highlights that mathematical programming is useful as a tool to connect predictions with decision-making constraints, rather than as an all-encompassing model for learning.
mathematical programminglinear programmingmixed-integer programmingoptimizationmachine learningartificial intelligencedistribution shiftmulti-objective optimizationtractabilityapproximation guarantees
Authors
Chaosheng Dong
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.