Interval Algorithms for Finding the Minimal Root in a Set of Multiextremal One-Dimensional Nondifferentiable Functions

Abstract

Two problems arising very often in applications are considered. The first problem consists of finding the minimal root of an analytic one-dimensional function over a given interval. It is supposed that the objective function can be multiextremal and nondifferentiable. Three new algorithms based on interval analysis and branch-and-bound global optimization approaches are proposed for solving this problem. The novelty of the new algorithms is in improving the elimination criteria and the order in which interval and point evaluations are realized. The techniques introduced accelerate the search in comparison with interval analysis methods traditionally used for finding roots of equations. The second problem considered in the paper is a generalization of the first one and deals with the search for the minimal root in a set of multiextremal and nondifferentiable functions. The methods proposed for solving the first problem are generalized for solving the second. The main idea is to use the information obtained from any of the functions to reduce the search domain associated with all the functions. Numerical experiments carried out on a wide set of test functions demonstrate a quite satisfactory performance of the new algorithms in both cases.