A geometric characterization of unbounded integer cubic optimization problems via thin rays

Abstract

We study geometric characterizations of unbounded integer polynomial optimization problems. Unboundedness along a ray characterizes unbounded integer linear and quadratic optimization problems with rational coefficients. We show that this is no longer true in degree three, already in dimension three, in contrast with the continuous setting, where rays certify unboundedness up to degree three. To recover a ray-based certificate, we introduce thin rays, which are rays with an arbitrarily small neighborhood. Our main result is that thin rays characterize unboundedness for integer cubic optimization problems over arbitrary rational polyhedra, in every dimension. As a special case, we obtain a characterization of unbounded integer quadratic optimization problems with irrational coefficients, a setting in which thin rays are again necessary. Degree three is also the threshold for certificates of this type: from degree four on, no ray certifies unboundedness, even if we allow a neighborhood of any width around it, already in dimension two and with rational coefficients. Together, these results give a complete picture of when unboundedness of an integer polynomial optimization problem is certified by a ray, by a thin ray, or by neither.