Fast Global Optimization of Di cult

The minimization of the potential energy function of Lennard-Jones atomic clusters has attracted much theoretical as well as computational research in recent years. One reason for this is the practical importance of discovering low-energy conngurations of clusters of atoms, in view of applications and extensions to molecular conformation research; another reason of the success of Lennard Jones minimization in the global optimization literature is the fact that this is an extremely easy-to-state problem, yet it poses enormous diiculties for any unbiased global optimization algorithm. In this paper we propose a computational strategy which allowed us to rediscover most putative global optima known in the literature for clusters of up to 80 atoms and for other larger clusters, including the most diicult cluster conformations. The main feature of the proposed approach is the deenition of a special purpose local optimization procedure aimed at enlarging the region of attraction of the best atomic conngurations. This eeect is attained by performing rst an optimization of a modiied potential function and using the resulting local optimum as a starting point for local optimization of the Lennard Jones potential. Extensive numerical experimentation is presented and discussed, from which it can be immediately inferred that the approach presented in this paper is extremely eecient when applied to the most challenging cluster conformations. Some attempts have also been carried out on larger clusters, which resulted in the discovery of the diicult optimum for the 102 atom cluster and for the very recently discovered new putative optimum for the 98 atom cluster.