Generalized Pattern Search Methods For Optimization Problems

Now,many applications of algorithms that do not employ the explicit derivatives information of the optimization problems have been developed more and more.Generalized pattern search methods is the main work in this paper,which is a particular subset of direct search methods that neither compute nor approximate any derivatives,and consequently do not need to enforce explicitly a notion of sufficient decrease to guarantee the convergence. Thus,this class of methods is suitable for problems when the objectives are complicated or their derivatives are difficult to compute.The main work in this paper has two parts:first,we continue to analyze the local convergence properties of pattern search method established in[8]and then we propose the generalized pattern search algorithm for finite minimax problems.The main results,obtained in this paper,may be summarized as follows:1.Chapter 2 studies the local convergence properties of the method established in[8],which is based on projecting the constrained problem onto the null space of the constraint matrix and therefore simplifying the complexity.In the procedure of proof,we always suppose the objective function has the property of second-order continuously differentiability in the neighbor of an isolated local minimizer.Some added constraints are enforced on the set of directions and the step-length control parameter which can provide a reliable asymptotic measure of first-order stability,and we analyze the behavior of pattern search in the neighborhood of an isolated local minimizer using this stationary measure.2.Chapter 3 constructs a new pattern search algorithm for minimax problems and gives the analytical results of the global convergence.We convert the original problem into an approximate smooth one by using a smoothing parameter,and then propose the pattern search method for the approximate smooth problem,and therefore get the stationary point of the original problem under some additional conditions,which method solve the non-smoothness of the original problem and get the global convergence results despite the explicit information concerning the first or second-order gradients.