A Large-Scale Stochastic-Perturbation Global Optimization Method for Molecular Cluster problems

We describe a new stochastic global optimization algorithm, and its application to a class of problems in molecular chemistry. The global optimization problem we consider is to find the lowest minimizer of a nonlinear function f that may contain multiple local minimizers, restricted to a bounded domain where f is assumed to be twice continuously differentiable. The two applications described in this paper both involve the determination of the structure of clusters of atoms or molecules, but each application uses a different potential energy function. The first potential is given by the sum of the pairwise interactions between atoms described by the Lennard-Jones function, and the second is the empirical water dimer potential energy surface function (RWK2-M) described in [10]. Problems in determining molecular structure lead to optimization problems because the naturally occurring structure usually minimizes the potential energy of the system. These problems become global optimization problems because typically such functions have very many local minimizers. Problems in molecular chemistry that are of practical interest may involve hundreds or thousands of atoms in three-dimensional space; therefore it is important to develop methods that are capable of solving large-dimensional problems. We present results for problems with up to 228 variables (76 atoms) using the Lennard-Jones potential, and 288 variables (32 water molecules) using the water potential, all of which compare favorably with other work on these test cases. More importantly, the techniques we develop, which combine computations in the full parameter space with phases that concentrate on smalldimensional subproblems within the overall problem, are designed to be applicable to a wide variety of structured large-scale global optimization problems. In particular, the framework for the technique we describe has been successfully applied to some protein conformation problems. The new algorithm is related to the stochastic methods of [32]. These stochastic methods combine random sampling over a specified domain space and local minimizations from selected sample points. These methods appear to be at least as efficient as