Geoffrion's theorem beyond finiteness and rationality

2025-10-12

AI Summary

This paper addresses the applicability of Geoffrion’s theorem under nonstandard feasible regions: when the feasible set is infinite and not representable by a finite system of rational linear constraints, the classical conclusion may fail. To overcome the traditional reliance on finiteness and rational linear constraints, the authors employ tools from convex analysis, topology, and mathematical programming—constructing explicit counterexamples to elucidate the failure mechanism. They then derive a new set of sufficient conditions that accommodate infinite domains and nonlinearly representable feasible sets, thereby preserving theoretical guarantees on the quality of Lagrangian relaxation bounds. The results rigorously extend the scope of Geoffrion’s theorem and establish a solid theoretical foundation for dual bound analysis in nonstandard integer programming problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible solutions is finite or described by rational linear constraints. However, we show through concrete examples that the conclusion of Geoffrion's theorem does not necessarily hold when this condition is dropped. We then provide sufficient conditions ensuring the validity of the result even when the feasible set is not finite and cannot be described using finitely-many linear constraints.

Problem

• Extends Geoffrion's theorem beyond finite rational constraints
• Identifies failure cases when feasibility conditions are relaxed
• Provides validity conditions for infinite non-rational feasible sets

Innovation

• Extends Geoffrion's theorem beyond finite sets
• Identifies conditions for non-finite feasible sets
• Ensures Lagrangian relaxation validity without rationality