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Solving clustered low-rank semidefinite programs arising from polynomial optimization

Nando Leijenhorst, David de Laat

TL;DR

This work develops a high-precision primal-dual interior-point method that exploits clustered low-rank semidefinite constraints arising from sums-of-squares formulations of polynomial optimization problems in discrete geometry. By integrating sampling-based representations, symmetry reduction, and a greedy procedure for selecting good sample points and bases, the approach achieves substantial speedups and improved numerical stability over prior solvers. The authors implement a Julia-based solver with fast matrix multiplication and demonstrate the method on two core applications: the three-point kissing-number bound and the binary sphere packing bound, obtaining a 28x speedup in the kissing-number case and extending bounds to higher degrees and dimensions. These advances enable solving previously intractable high-degree instances, yielding improved bounds in dimensions $11$--$23$ and providing new insights into conditioning and convergence for matrix polynomial programs via clustered low-rank structure.

Abstract

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions $11$ through $23$. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.

Solving clustered low-rank semidefinite programs arising from polynomial optimization

TL;DR

This work develops a high-precision primal-dual interior-point method that exploits clustered low-rank semidefinite constraints arising from sums-of-squares formulations of polynomial optimization problems in discrete geometry. By integrating sampling-based representations, symmetry reduction, and a greedy procedure for selecting good sample points and bases, the approach achieves substantial speedups and improved numerical stability over prior solvers. The authors implement a Julia-based solver with fast matrix multiplication and demonstrate the method on two core applications: the three-point kissing-number bound and the binary sphere packing bound, obtaining a 28x speedup in the kissing-number case and extending bounds to higher degrees and dimensions. These advances enable solving previously intractable high-degree instances, yielding improved bounds in dimensions -- and providing new insights into conditioning and convergence for matrix polynomial programs via clustered low-rank structure.

Abstract

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions through . The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.
Paper Structure (10 sections, 8 theorems, 69 equations, 6 figures, 1 table)

This paper contains 10 sections, 8 theorems, 69 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. A specialized interior point method
  3. Polynomial matrix programs
  4. Combining symmetry reduction and sampling
  5. Computing good sample points and bases
  6. Applications
  7. Kissing number problem
  8. Binary sphere packing
  9. Symmetric description of a symmetric semialgebraic set
  10. Nonexistence of an invariant minimal unisolvent set

Key Result

Theorem 3.1

Let $f \in \mathbb{R}[x]^{m \times m}$ and $G\subseteq \mathbb{R}[x]$ finite. Suppose $\mathcal{M}(G)$ is Archimedean. If $f \succ 0$ on $\mathcal{S}(G)$, then $f \in \mathcal{M}(G)$.

Figures (6)

  • Figure 6.1: The time needed to compute the three-point bound for the kissing number in dimension $n=4$ for several degrees $d$ on a linear scale, using SDPA-GMP and ClusteredLowRankSolver.
  • Figure 6.2: The time needed to compute the three-point bound for the kissing number in dimension $n=4$ for several degrees $d$ on a log-log scale, using SDPA-GMP and ClusteredLowRankSolver.
  • Figure 6.3: The binary sphere packing bound in dimension $n=2$ and Florian's bound.
  • Figure 6.4: Binary sphere packing upper bounds in dimension $n=8$.
  • Figure 6.5: The binary sphere packing bound in dimension $24$. The dotted line is the maximum density of single sphere packings and the dashed line the optimal density when $r$ tends to $0$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 3.1: scherer_matrix_2006klep_pure_2010
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 3 more