Research On The Improved Nonmonotone Trust Region Algorithm

Optimization problems exist in many fields such as information engineering,social economy and so on.Trust region method is an important method in optimization algorithm,and it is easier to establish convergence and robustness.The main research content of this paper is the improved trust-region method for solving general unconstrained optimization problems,nonlinear equations and nonlinear least squares problems.In the Chapter 1,we introduce the background and significance of the problem firstly,and the status of trust region algorithm in the optimization algorithm and the basic thought.Secondly,the trust region algorithm research progress at home and abroad are introduced,including the introduction of research problems,the origination and development of trust region algorithm,the solution of trust region subproblem,radius update,trust region algorithm combined with other techniques.Finally,the structure and research content of this paper are given.We display some related basic knowledge in the Chapter 2.Firstly,several common methods for solving unconstrained optimization problems are introduced,including conjugate gradient method,quasi-Newton method and gradient method in line search type method.Next,the basic structure of trust region algorithm for unconstrained optimization problem and nonlinear equations are given.In addition,the definitions of convergence and convergence rate are introduced.Considering the unconstrained optimization problems,an adaptive nonmonotone trust region algorithm is proposed in Chapter 3.The adaptive technique is introduced to update the radius of the trust region subproblem.At the same time,the nonmonotonic technique is introduced to adjust the trust region ratio in order to reduce the convergence rate of the traditional algorithm when the narrow valley phenomenon occurs.In addition,combining the nonmonotone line search strategy that can avoid resolving the trust region subproblem whenever a trial step is rejected.Under general assumptions,the global convergence and superlinear convergence rate of the algorithm are proved.Numerical experiments also indicate that the improved algorithm is feasible and effective.In Chapter 4,an improved nonmonotone trust region algorithm is proposed for solving nonlinear equations.The algorithm not only uses the convex combination of the maximum function value and the current function value to construct a nonmonotone term,but also combines the line search strategy to solve the trust-region subproblem when the trial step fails,so as to reduce the computational complexity of the algorithm.And the global convergence property and the nearly Q-cubic convergence rate of the algorithm are verified under certain conditions.Finally,the preliminary numerical experiments show that the algorithm is feasible.At the end of the paper,the conclusion is made and the future research direction is prospected.