ClusterAnalysisUsing Seed Points and Density-DeterminedHyperspheres as an Aid toGlobalOptimization

A model forfinding thelocal optima ofamultimodal function defined inaregion A cR'isproposed. Themethod uses a local optimizer which isstarted fromanumber ofpoints sampled in A.Inorder toreduce thenumber offunction evaluations needed to reach thelocal optima, theparallel local search processes arestopped repeatedly, theworking points clustered, andareduced number of processes fromeach cluster resumed. Adirect nonhierarchical cluster analysis technique ispresented. Thedissimilarity measure used isthe Euclidean distance between points. Clusters aregrownfromseed points. Thenumber ofrequired distance evaluations isless thanor equal toc(n - 1), where nisthenumber ofpoints andcisthenumber ofclusters arrived at. Thresholds aredetermined bythepoint density inabodywhich inturnisdetermined bythegiven points. The covariance matrix isdiagonalized, andadecision onthedimension- ality ofthespace containing thepoints canbemade. Thevolume of thebodyisproportional tothesquare rootoftheproduct ofthe corresponding eigenvalues. Theperformance oftheclustering analysis technique isillustrated. Itisdemonstrated that there exist classes ofglobal optimization problems forwhich theprobability of obtaining asolution isgreater fortheproposed modelthanfor multiple local optimizations. Someexperiences gained fromusing themodel arereported.