Pattern Search Methods for User-Provided Points: Application to Molecular Geometry Problems

This paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry. The goal is to find a configuration of a cluster or a molecule with minimum total energy. The minimization problems in this class of molecular geometry problems have no constraints, and the objective function is smooth. The difficulties arise from the existence of several local minima and, especially, from the expensive function evaluation (total energy) and the possible nonavailability of first-order derivatives. We introduce a pattern search approach that attempts to exploit the physical nature of the problem by using energy lowering geometrical transformations and to take advantage of parallelism without the use of derivatives. Numerical results for a particular instance of this new class of pattern search methods are presented, showing the promise of our approach. The new pattern search methods can be used in any other context where there is a user-provided scheme to generate points leading to a potential objective function decrease.