Algorithm Description

The objective of the cluster optimization problem is to minimize the total potential energy and thereby determine the lowest energy structure. As stated before we have encoded the positions of the atoms in cartesian coordinates using a 5 bit binary string for each coordinate. The initial population is generated randomly, with each coordinate sampled uniformly between $\left(0, \sigma\sqrt[3]{6n}\right)$, were $\sigma = 2.0591$. These structures are then relaxed using a local search method, in our case a Nelder-Mead simplex method. Nelder-Mead simplex requires $m+1$ initial points for an $m$ variable problem. The initial points for every individual in a population is generated by adding a uniform random variabe between $\left(0,0.1*\sigma\right)$ to one of variables. For eg. if we are solving for a 4 atom cluster, then we have 12 variables. Therefore we need 13 initial clusters for the simplex method. A given individual would account for one initial value, and the rest 12 are generated by adding a uniform random value to one of the 12 variables of the given individual.

Unlike Iwamatsu [10] who ran the simplex algorithm till a tolerance of 10^-6 was reached, we just run it till a tolerance of 10^-3 is reached. There are two ways of incorporating the solution obtained by the local search, one is just to assign the fitness of the local search to individual without changing the structure (Baldwinian approach) and the other is the assign both the fitness as well as the structure to the individual (Lamarckian). A rule of thumb states that Baldwinian approach should be applied for 95% of the time and Lamarckian for the rest 5%. However the present case seems to be an exception, because a preliminary study indicated that a fully Lamarckian approach is superior to the 95% Baldwinian approach. Therefore in the remainder of the study we have used a 100% Lamarckian approach.

These relaxed structures are then subjected to a tournament selection without replacement. i.e., a given number of individuals are selected from the current generation and the individual with minimum potential energy is transferred to a mating-pool. This procedure is repeated till the mating pool has exactly N_p individuals. In the present study we used tournament sizes of 8, 16, and 20. All the results reported in the present study, unless otherwise mentioned, are averages of 25 independent runs. The Nelder-Mead simplex algorithm was adopted from [16].

The convergence criteria used were as follows (1) the variance of fitness of the population is less than 0.1, or (2) the variance of bit values of the population is less than 0.1, whichever occurs first. The minimum population size, convergence time and the number of function evaluations were evaluated when the ECGA failed at most once out of 25 runs. Therefore the reliability of the algorithm was set to 96%. These criteria are much more stringent than those existing in literature that just require one out of 10 runs (reliablity of 10%) to reach the global minima.