Research On Nonmonotone Trust Region Algorithms For Nonlinear Optimization

Trust region algorithm is a class of very important numerical algorithms for non-linear programming problems. In recent years, because of trust region algorithm ofstability and robustness, there are many researchers pay more attention to this kind ofmethods, which become a hot topic in optimization field.In this paper, there are four results as following.In chapter 2, a new trust region method is proposed for solving unconstrainedoptimization problems. Trail step size d~k is always generated in trust region of thesubproblem, and which updated at every iteration. When d~k is not accepted, we canuse nonmonotone line search to solve subproblem. Numerical results show the methodis effective in minimizing unconstrained optimization problems.In chapter 3, we propose a new filter-trust region algorithm for unconstrainedoptimization problems. A new filter technique is introduced to solve the problem whenthe direction d~k is not accepted. The theoretical analysis shows that the algorithm isnot only global convergence but also R-linearly convergence, and even super linearlyconvergence can be obtained under some suitable conditions. Numerical results showthe method is effective in minimizing unconstrained optimization problems.In chapter 4, a conic trust region algorithm is proposed for unconstrained opti-mization problems. The method can be regarded as a combination of nonmonotoneline search technique, truncated Quasi-Newton method and conic trust region method.When trail step is not accepted, we can use nonmonotone line search rules for a suitablestep length, then generate next iterative point. It need not resolve the conic trust re-gion subproblem. The theoretical analysis shows that the algorithm is not only globalconvergence but super linearly convergence under some suitable conditions. Numer-ical results show the algorithm is effective in minimizing unconstrained optimizationproblems.In chapter 5, inequality constrained optimization problems are discussed, based ona combination technique of a trust region method and an SQP method, a new feasiblealgorithm is proposed. A compression technique is used such that search direction isfeasible for QP subproblem at every iteration. We use high order revised directionto avoid Maratos e?ect. Under some suitable conditions, the global and superlinearconvergence can be induced.