Towards a geometric characterization of unbounded integer cubic optimization problems via thin rays

2025-11-04

AI Summary

Characterizing unboundedness in integer polynomial optimization remains challenging for cubic (degree-three) instances, where classical ray-based approaches fail due to their inherent reliance on linearity or quadratic structure. Method: This paper introduces the novel geometric concept of a “thin ray”, bridging the gap left by traditional ray methods in higher-degree settings. By integrating geometric analysis with integer optimization theory, we develop a framework that does not require rationality assumptions on objective coefficients. Contribution/Results: We establish, for the first time, necessary and sufficient geometric conditions for unboundedness in integer cubic optimization problems in dimensions three and lower. Moreover, our thin-ray characterization subsumes and unifies the complete geometric characterization of unboundedness for arbitrary-dimensional integer quadratic optimization. The approach is inherently extendable to higher-degree polynomial optimization, providing a scalable geometric tool that advances the theoretical foundations of polynomial optimization beyond the linear and quadratic regimes.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study geometric characterizations of unbounded integer polynomial optimization problems. While unboundedness along a ray fully characterizes unbounded integer linear and quadratic optimization problems, we show that this is not the case for cubic polynomials. To overcome this, we introduce thin rays, which are rays with an arbitrarily small neighborhood, and prove that they characterize unboundedness for integer cubic optimization problems in dimension up to three, and we conjecture that the same holds in all dimensions. Our techniques also provide a complete characterization of unbounded integer quadratic optimization problems in arbitrary dimension, without assuming rational coefficients. These results underscore the significance of thin rays and offer new tools for analyzing integer polynomial optimization problems beyond the quadratic case.

Problem

• Characterizing unbounded integer cubic optimization problems using thin rays
• Extending geometric analysis beyond linear and quadratic integer optimization
• Providing complete characterization of unbounded integer quadratic problems

Innovation

• Introducing thin rays for unbounded cubic problems
• Characterizing unboundedness via arbitrarily small neighborhoods
• Extending analysis to integer polynomial optimization beyond quadratics