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

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 epsilon, the exact gradient of the smoothed function, U-epsilon, is a simple three-point finite difference of U. When epsilon is very large, the gradient of U-epsilon appears to be a "bad derivative" of U. Because the gradient of U-epsilon is a simple function of U, minimization on the smoothed surface requires no explicit calculation or differentiation of U-epsilon. 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. (C) 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1445-1455, 1998.