Exact semidefinite programming bounds for packing problems

Maria Dostert, David de Laat, Philippe Moustrou

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)

Abstract

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. This algorithm does not require the solution to be strictly feasible and works for large problems. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the E8 root lattice is the unique optimal code with minimal angular distance π/3 on the hemisphere in R8, and we_prove that the three-point bound for the (3, 8, ϑ)-spherical code, where ϑ is such that cos ϑ = (2√2 - 1)/7, is sharp by rounding to Q[√2]. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.

Original languageEnglish
Pages (from-to)1433-1458
Number of pages26
JournalSIAM Journal on Optimization
Volume31
Issue number2
DOIs
Publication statusPublished - 2021

Keywords

  • Hybrid numeric-symbolic algorithm
  • Packing problems
  • Semidefinite programming
  • Spherical codes

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