A Point Balance Algorithm for the Spherical Code Problem

Authors:
Hong-Xuan Huang;Panos M. Pardalos;Zuo-Jun Shen
Affiliations:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China/;Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA (e-mail: pardalos@cao.ise.ufl.edu);Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA
Venue:
Journal of Global Optimization
Year:
2001

Citing 5
Cited 1

Sphere-packings, lattices, and groups

Sphere-packings, lattices, and groups
Complexity of Bezout's theorem III: condition number and packing

Journal of Complexity - Festschrift for Joseph F. Traub, Part 1
New results in the packing of equal circles in a square

Discrete Mathematics
On the Convergence of Pattern Search Algorithms

SIAM Journal on Optimization
Pattern Search Algorithms for Bound Constrained Minimization

SIAM Journal on Optimization

Equivalent formulations and necessary optimality conditions for the Lennard–Jones problem

Journal of Global Optimization

Quantified Score

Hi-index	0.00

Visualization

Abstract

The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced spherical code is introduced and a new approach, the Point Balance Algorithm (PBA), is presented to search for a 1-balanced spherical code. Since an optimal solution of the SC problem (an extremal spherical code) must be a 1-balanced spherical code, PBA can be applied easily to search for an extremal spherical code. In addition, given a certain criterion, PBA can generate efficiently an approximate optimal spherical code on a sphere in the n-dimensional space Rn. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional space are also reported. Finally, an open question about the geometry of Fekete points on the unit sphere in the 3-dimensional space is posed.