Abstract
The purpose of this chapter is to present a two-level random search method for unconstrained optimization and the corresponding algorithm. The idea of the algorithm is to randomly generate a number of trial points in some domains at two levels. At the first level, a number of trial points are generated around the initial point, where the minimizing function is evaluated. At the second level, another number of local trial points are generated around each trial point, where the minimizing function is evaluated again. The algorithm consists of a number of rules for replacing the trial points with local trial points and for generating some new trial points to get a point where the function value is smaller. Some details of the algorithm are developed and discussed, concerning the number of trial points, the number of local trial points, the bounds of the domains where these trial points are generated, the reduction of the bounds of these domains, the reduction of the trial points, middle points, the local character of searching, the finding of the minimum points, and the line search used for accelerating the algorithm. The numerical examples illustrate the characteristics and the performances of the algorithm. At the same time, some open problems are identified.
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