| Unconstrained optimization problem is a very important class of prob-lems in the field of nonlinear optimization.There are many such problems in real life.As trust region method has nice convergence,it plays an impor-tant role in the area of nonlinear optimization and is among efficient methods for solving unconstrained optimization problem.This thesis mainly forces on adaptive trust region algorithms to solve smooth and nonsmooth uncon-strained optimization problems.The main achievements are shown as fol-lows:(1)An adaptive trust region algorithm for solving smooth unconstrained optimization problems are studied.Firstly,a modified adaptive trust region method is presented,where the trust region radius uses the first order gradi-ent information of the function.Under suitable conditions,the global conver-gence and the local superlinear convergence of the proposed algorithm are analyzed theoretically.Finally,the proposed algorithm compared with some existing algorithms,the preliminary numerical results indicate that the pro-posed algorithm is effective for solving smooth unconstrained optimization problems.(2)An adaptive trust region algorithm for solving nonsmooth uncon-strained optimization problems are considered.By way of the Moreau-Yosida regularization,a modified adaptive trust region method is presented.The pro-posed algorithm combines a modified secant equation with the BFGS updat-ed formula and an adaptive trust region radius,where the new trust region radius makes use of not only the function information but also the gradient information.Under suitable conditions,the global convergence and the local superlinear convergence of the proposed algorithm are proved.Moreover,the performance of the proposed algorithm is verified by testing some problems in numerical experiments,and compared with some algorithms.the prelim-inary numerical results reveal that the proposed algorithm is well promising to solve nonsmooth unconstrained optimization problems. |
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