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 geometry molecular 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 non-availability 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 with 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 an user-provided scheme to generate points leading to potential objective function decrease.
Preprint 00-20 Department of Mathematics, University of Coimbra, Portugal September 2000, Revised May 2002