Global optimization using bad derivatives: Derivative-free method for molecular energy minimization

Abstract

A general method designed to isolate the global minimum of a multidimensional objective function with multiple minima is presented. The algorithm exploits an integral “coarse-graining” transformation of the objective function, U, into a smoothed function with few minima. When the coarse-graining is defined over a cubic neighborhood of length scale ϵ, the exact gradient of the smoothed function, 𝒰ϵ, is a simple three-point finite difference of U. When ϵ is very large, the gradient of 𝒰ϵ appears to be a “bad derivative” of U. Because the gradient of 𝒰ϵ is a simple function of U, minimization on the smoothed surface requires no explicit calculation or differentiation of 𝒰ϵ. The minimization method is “derivative-free” and may be applied to optimization problems involving functions that are not smooth or differentiable. Generalization to functions in high-dimensional space is straightforward. In the context of molecular conformational optimization, the method may be used to minimize the potential energy or, preferably, to maximize the Boltzmann probability function. The algorithm is applied to conformational optimization of a model potential, Lennard–Jones atomic clusters, and a tetrapeptide. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1445–1455, 1998