Duality theory in linear optimization and its extensions - formally verified

2024-09-12

arXiv.org

AI Summary

The formal foundations of linear optimization duality theory remain underdeveloped, particularly regarding rigorous, machine-checkable verification. Method: We develop a sound and extensible, machine-verifiable framework in Lean 4, leveraging dependent types and algebraic structure modeling to formalize Farkas’ lemma and its variants over general ordered fields. Crucially, we extend duality theory to generalized linear inequality systems with coefficients in the extended reals (including ±∞), rigorously constructing the extended real number system and its order structure. Contributions: (1) The first complete, machine-checked formalization in Lean 4 of multiple Farkas-type theorems, together with precise necessary and sufficient conditions over arbitrary ordered fields; (2) The first mathematical modeling and formal verification of duality theory accommodating infinite coefficients; (3) A foundational, highly reliable basis for verifying optimization algorithms and building trustworthy mathematical libraries.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take ``infinite values''.

Problem

• Formally verify Farkas-like theorems in Lean 4
• Extend duality theory to infinite coefficients
• Analyze linear inequalities over ordered fields

Innovation

• Formally verified Farkas-like theorems
• Extended duality theory with Lean 4
• Infinite coefficients in linear optimization