GLOBAL OPTIMIZATION USING BAD DERIVATIVES - DERIVATIVE-FREE METHOD FOR MOLECULAR-ENERGY MINIMIZATION

A general method designed to isolate the global minimum of a multidime nsional objective function with multiple minima is presented. The algo rithm exploits an integral ''coarse-graining'' transformation of the o bjective function, U, into a smoothed function with few minima. When t he coarse-graining is defined over a cubic neighborhood of length scal e epsilon, the exact gradient of the smoothed function, U-epsilon, is a simple three-point finite difference of U. When epsilon is very larg e, 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, minimizat ion on the smoothed surface requires no explicit calculation or differ entiation 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 conf ormational optimization, the method may be used to minimize the potent ial energy or, preferably, to maximize the Boltzmann probability funct ion. The algorithm is applied to conformational optimization of a mode l potential, Lennard-Jones atomic clusters, and a tetrapeptide. (C) 19 98 John Wiley & Sons, Inc. J Comput Chem 19: 1445-1455, 1998.