Cluster Analysis Using Seed Points and Density-Determined Hyperspheres as an Aid to Global Optimization

A model for finding the local optima of a multimodal function defined in a region A ⊂ Rn is proposed. The method uses a local optimizer which is started from a number of points sampled in A. In order to reduce the number of function evaluations needed to reach the local optima, the parallel local search processes are stopped repeatedly, the working points clustered, and a reduced number of processes from each cluster resumed. A direct nonhierarchical cluster analysis technique is presented. The dissimilarity measure used is the Euclidean distance between points. Clusters are grown from seed points. The number of required distance evaluations is less than or equal to c(n-1), where n is the number of points and c is the number of clusters arrived at. Thresholds are determined by the point density in a body which in turn is determined by the given points. The covariance matrix is diagonalized, and a decision on the dimensionality of the space containing the points can be made. The volume of the body is proportional to the square root of the product of the corresponding eigenvalues. The performance of the clustering analysis technique is illustrated. It is demonstrated that there exist classes of global optimization problems for which the probability of obtaining a solution is greater for the proposed model than for multiple local optimizations. Some experiences gained from using the model are reported.