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

✍ Scribed by Andricioaei, Ioan; Straub, John E.

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 247 KB
• Volume: 19
• Category: Article
• ISSN: 0192-8651
• DOI: 10.1002/(sici)1096-987x(199810)19:13<1445::aid-jcc2>3.0.co;2-q

✦ Synopsis

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 coarsegraining is defined over a cubic neighborhood of length scale ⑀ , the exact gradient of the smoothed function, U U , is a simple three-point finite difference of ⑀ U. When ⑀ is very large, the gradient of U U appears to be a ''bad derivative'' of ⑀ U. Because the gradient of U U is a simple function of U, minimization on the ⑀ smoothed surface requires no explicit calculation or differentiation of U U . 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. ᮊ 1998