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

Alberto Del Pia

TL;DR

The paper investigates when unboundedness of integer cubic optimization can be inferred from geometric directions. It shows that unlike linear and quadratic cases, cubic objectives require the notion of thin rays—rays together with arbitrarily small neighborhoods—to characterize unboundedness in dimensions up to $3$, and it proves a main theorem confirming this for $n\le 3$ while conjecturing general validity. It also provides a complete, dimension-agnostic characterization for unbounded integer quadratic problems without assuming rational coefficients, using tensor-based and projection techniques. The work advances the understanding of unboundedness in integer polynomial optimization beyond the quadratic case and introduces methods (tensor contractions, projection along recession directions, and Dirichlet-type approximations on rational polyhedra) that may extend to higher degrees and dimensions. The results highlight the essential role of thin rays and offer new tools for analyzing unboundedness and growth in integer polynomial optimization.

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.

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

TL;DR

The paper investigates when unboundedness of integer cubic optimization can be inferred from geometric directions. It shows that unlike linear and quadratic cases, cubic objectives require the notion of thin rays—rays together with arbitrarily small neighborhoods—to characterize unboundedness in dimensions up to , and it proves a main theorem confirming this for while conjecturing general validity. It also provides a complete, dimension-agnostic characterization for unbounded integer quadratic problems without assuming rational coefficients, using tensor-based and projection techniques. The work advances the understanding of unboundedness in integer polynomial optimization beyond the quadratic case and introduces methods (tensor contractions, projection along recession directions, and Dirichlet-type approximations on rational polyhedra) that may extend to higher degrees and dimensions. The results highlight the essential role of thin rays and offer new tools for analyzing unboundedness and growth in integer polynomial optimization.

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.

Paper Structure

This paper contains 39 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. On the rationality of the polynomial coefficients.
  3. Linear, bilinear, and trilinear forms
  4. Proof of Theorem \ref{['th main']}
  5. Proof of Proposition \ref{['prop int ray']}
  6. First projection.
  7. Second projection.
  8. Proof of Proposition \ref{['prop cubic example']}
  9. Proof of Observation \ref{['obs f notation']}
  10. Remaining arguments for the proof of Theorem \ref{['th main']}
  11. Proofs of claims in Proposition \ref{['prop int ray']}
  12. Proof of Claim \ref{['claim pointed']}
  13. Proof of Claim \ref{['claim integral']}
  14. Proof of Claim \ref{['claim H']}
  15. Proof of Claim \ref{['claim rec full']}
  16. ...and 24 more sections

Key Result

theorem 1

Let $\mathcal{P} \subseteq \mathbb R^n$ be a rational polyhedron and let $f \colon \mathbb R^n \to \mathbb R$ be a linear or quadratic function with rational coefficients. Then $f$ is unbounded below on $\mathcal{P} \cap \mathbb Z^n$, if and only if there exists a ray $\mathcal{R}(y, d)$ of $\mathca

Theorems & Definitions (16)

  • theorem 1
  • proposition 1
  • theorem 2
  • proposition 2
  • theorem 3
  • proposition 3
  • proposition 4
  • lemma 1
  • proof
  • proof
  • ...and 6 more