Charged balls method for solving some computational geometry problems

Abstract

The concept of replacing the original stationary optimization problem with a nonstationary mechanical system that tends to the position of equilibrium, which coincides with the solution of the initial problem, allows us to build effective numerical algorithms. First, differential equations of movement should be derived. After that we can get to the difference scheme and thus define iterative computational algorithm. There is a wide class of optimization methods built that way. One of the most known representatives of this class is heavy ball method. Such type of algorithms always have parameters which highly affect the rate of convergence. In this paper we propose and investigate charged balls method which is the other representative of this class. It is a new effective optimization method, that allows to solve some computational geometry problems. We consider the problem of orthogonal projection of a point onto a convex closed set with smooth boundary, and the problem of finding the minimum distance between two such sets. Theorems of convergence and estimates for the rate of convergence are obtained. Numerical examples illustrating the work of the proposed algorithms are given. Refs 8. Tables 2.