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

TL;DR

This paper introduces thin rays as a new geometric tool to characterize unboundedness in integer cubic optimization problems, extending known results from quadratic cases and providing insights up to three dimensions.

Contribution

It proposes thin rays to characterize unboundedness in integer cubic problems and fully characterizes unbounded quadratic problems without rationality assumptions.

Findings

Thin rays characterize unboundedness in cubic problems up to dimension three.

Complete characterization of unbounded quadratic problems in arbitrary dimensions.

Demonstrates limitations of rays for cubic problems beyond dimension three.

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.