Global optimization and structural analysis of Coulomb and logarithmic potentials on the unit sphere using a population-based heuristic approach

Abstract

Global optimization of high-dimensional non-convex functions arises in many important applications and researches. In this paper, we focus on the minimum energy configuration problem on the unit sphere, whose goal is to determine the minimum-energy configuration of N particles interacting via the standard Coulomb potential or the discrete logarithmic potential on the unit sphere. This is a classic global optimization problem with many important applications in mathematics, physics, biology and chemistry, and is one of the 18 great mathematical problems for the twenty-first century listed by the Fields Medal winner (Smale, 1998). To determine the minimum-energy configuration for these two potential functions, we propose a population-based global optimization algorithm that combines a two-phase local optimization method with early-stopping strategy, a random perturbation operator, a distance-based population update strategy using a novel distance function to measure the difference between solutions, and a population rebuilding method. Computational results show that the proposed algorithm is very competitive compared to the best known results in the literature. In particular, it improves the best-known result for one instance of the classic Thomson problem (i.e., N=360) widely studied in the literature. The configurations of instances with up to N=500 particles are systematically optimized for the first time by the proposed algorithm for the discrete logarithmic potential. The comparative study shows that the putative optimal configurations have a high similarity for the Coulomb and logarithmic potentials and that the defects play an important role in reducing the energy of configuration. The energies of particles on the putative optimal configurations are analyzed for the studied logarithmic potential.