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

arXiv: https://arxiv.org/abs/2609.07254 · HTML: https://mlsanity.com/paper/2609.07254/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 34 references that are in the map.
- First posted on arXiv 7 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Mixture-of-Experts with Expert Choice Routing](https://mlsanity.com/paper/2202.09368/) (2022)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery](https://mlsanity.com/paper/2405.09783/) (2024)
- [Accelerating Scientific Discovery with Autonomous Goal-evolving Agents](https://mlsanity.com/paper/2512.21782/) (2025)
- [Robust LLM Alignment via Distributionally Robust Direct Preference Optimization](https://mlsanity.com/paper/2502.01930/) (2025)
- [Generalized Inverse Optimization through Online Learning](https://mlsanity.com/paper/1810.01920/) (2018)
- [Threading Keyframe with Narratives: MLLMs as Strong Long Video Comprehenders](https://mlsanity.com/paper/2505.24158/) (2025)
- [Automated Composition of Agents: A Knapsack Approach for Agentic Component Selection](https://mlsanity.com/paper/2510.16499/) (2025)
- [Self-Correcting RAG: Enhancing Faithfulness via MMKP Context Selection and NLI-Guided MCTS](https://mlsanity.com/paper/2604.10734/) (2026)
- [A First-order Generative Bilevel Optimization Framework for Diffusion Models](https://mlsanity.com/paper/2502.08808/) (2025)

## Abstract (arXiv)

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.