[Paper Review] Mathematical optimization for packing problems

Fernando Mário de Oliveira Filho, Frank Vallentin|arXiv (Cornell University)|Mar 5, 2014

Computational Geometry and Mesh Generation12 references3 citations

TL;DR

This paper presents a unified framework using semidefinite optimization and harmonic analysis to derive upper bounds for geometric packing problems, including sphere packings, binary sphere packings, and convex body packings. By extending the Lovász theta number to continuous spaces via Fourier analysis and sums-of-squares representations, it enables numerical computation of tighter density bounds, with successful applications yielding new upper bounds for pentagon packings and binary sphere packings in low dimensions.

ABSTRACT

During the last few years several new results on packing problems were obtained using a blend of tools from semidefinite optimization, polynomial optimization, and harmonic analysis. We survey some of these results and the techniques involved, concentrating on geometric packing problems such as the sphere-packing problem or the problem of packing regular tetrahedra in R^3.

Motivation & Objective

• To develop a systematic method for computing upper bounds on packing densities in Euclidean space using optimization techniques.
• To extend combinatorial optimization tools like the Lovász theta number to continuous geometric packing problems.
• To address the computational challenge of applying these methods to non-compact spaces like Euclidean space.
• To provide a practical and numerically stable formulation of polynomial optimization problems arising in packing bounds.

Proposed method

• Formulate the packing density problem as a semidefinite program using polynomial optimization and sums-of-squares representations.
• Employ Fourier analysis and spherical harmonics to encode symmetry and constraints in continuous packing problems.
• Use a transformed monomial basis, specifically $ P_k(t) = \mu_k^{-1} L_k^{n/2-1}(2\pi t) $, to improve numerical stability in solving the SDP.
• Introduce matrix-valued kernels and constraints involving positive semidefinite matrices $ Q, R, S $ to model correlation functions.
• Apply the Lasserre hierarchy conceptually to incorporate higher-order (k-point) correlation functions beyond 2-point functions.
• Leverage the duality between optimization and harmonic analysis to derive bounds that are both theoretically sound and computationally accessible.

Experimental results

Research questions

• RQ1 How can the Lovász theta number be generalized to continuous geometric packing problems in Euclidean space?
• RQ2 What is the role of harmonic analysis and Fourier transforms in formulating tight upper bounds for packing densities?
• RQ3 How can polynomial optimization with sums-of-squares constraints be made numerically stable for high-degree polynomials?
• RQ4 Can higher-order correlation functions (e.g., 3-point or more) improve the tightness of packing density bounds?
• RQ5 What basis choices in polynomial representation yield the most efficient and accurate solutions in semidefinite programming for packing problems?

Key findings

• The paper successfully extends the Lovász theta number framework to geometric packing problems, enabling a unified approach to upper bounding packing densities.
• A new basis $ P_k(t) = \mu_k^{-1} L_k^{n/2-1}(2\pi t) $ is identified as highly effective for numerical stability in solving the resulting semidefinite programs.
• The method yields a first upper bound of 0.98 for pentagon packing density, compared to the best known lower bound of 0.92, indicating strong potential for future improvements.
• The approach provides new upper bounds for binary sphere packings in dimensions 2 through 5, extending prior work by De Laat, Oliveira, and Vallentin.
• The framework reveals that higher-order correlation functions (e.g., 3-point) can theoretically tighten bounds, though at the cost of rapidly increasing computational complexity.
• The authors demonstrate that the use of matrix-valued Fourier kernels and semidefinite constraints allows for a systematic treatment of non-compact packing problems.