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Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones

Jesús A. De Loera, Brittney Marsters, Luze Xu, Shixuan Zhang

TL;DR

The paper develops a framework to study lattice points in arbitrary convex cones by introducing $(R,G)$-finite generation, where a finitely generated group $G\le\mathrm{GL}(N,\mathbb{Z})$ acts on the cone and its lattice points to produce a finite root set $R$ and a finite generating set. It proves that the semigroups of integer points in the PSD cone and, for $3\le n\le 10$, the SOC cone are $(R,G)$-finitely generated, with explicit $R$ and $G$ in each case, while providing a counterexample showing not all conical semigroups admit such finite generation. The results extend classical Hilbert-basis notions to nonpolyhedral cones and yield practical descriptions for Chvátal-Gomory cuts, total dual integrality, and the integer Carathéodory rank in this broader setting. The findings have implications for optimization on spectrahedra and conic integer programs, offering structural insights and algorithmic avenues via the group action. Overall, the work bridges geometry of numbers with conic optimization, revealing finite-generation phenomena in key cones and delineating limits of the approach.

Abstract

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones.

Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones

TL;DR

The paper develops a framework to study lattice points in arbitrary convex cones by introducing -finite generation, where a finitely generated group acts on the cone and its lattice points to produce a finite root set and a finite generating set. It proves that the semigroups of integer points in the PSD cone and, for , the SOC cone are -finitely generated, with explicit and in each case, while providing a counterexample showing not all conical semigroups admit such finite generation. The results extend classical Hilbert-basis notions to nonpolyhedral cones and yield practical descriptions for Chvátal-Gomory cuts, total dual integrality, and the integer Carathéodory rank in this broader setting. The findings have implications for optimization on spectrahedra and conic integer programs, offering structural insights and algorithmic avenues via the group action. Overall, the work bridges geometry of numbers with conic optimization, revealing finite-generation phenomena in key cones and delineating limits of the approach.

Abstract

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones.
Paper Structure (9 sections, 21 theorems, 74 equations)

This paper contains 9 sections, 21 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. The Positive Semidefinite (PSD) Cone
  3. Lemmas for $n\le 5$ and $n=6$
  4. Proof of Theorem \ref{['thm:PSD']}
  5. The Second-Order Cone (SOC)
  6. Sporadic Points of $\text{SOC}(n)\cap \mathbb{Z}^n$
  7. Proof of Theorem \ref{['thm:SOC']}
  8. Not All Conical Semigroups are $(R,G)$-Finitely Generated
  9. Applications: Total Dual Integrality, Chvátal-Gomory Closures, and Integer Rank

Key Result

Theorem 1

The conical semigroup of the cone of $n\times n$ positive semidefinite matrices, $\calS^n_+(\mathbb{Z})$, is $(R,G)$-finitely generated by $G\cong\mathrm{GL}(n,\mathbb{Z})$ where $G$ acts on $X\in\calS^n_+(\mathbb{Z})$ by $X\mapsto UXU^\mathsf{T}$ for each $U\in\mathrm{GL}(n,\mathbb{Z})$, and by $R$

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Proposition 6
  • proof
  • ...and 37 more