The New Cone Model Trust Region Algorithm

Trust region algorithm based on quadratic model has become an important class of numerical calculation methods because of its strong adaptability and convergence of optimization algorithms. But for strong non-quadratic form and the sharp change in curvature function, the quadratic model seems to unable to solve it. In recent years, the conic model for the study of trust region algorithm has aroused widespread concern. Especially, the trust region algorithm based on new conic model was proposed in 2005,it is not only made up for the defects of the quadratic model turst region algorithm,but also broken through the limitations of traditional conic model trust region algorithm on solving optimal value in one side of the plane. Therefore, this article mainly studies in solving sub-problem, the structure of the algorithm frame and the improvement. Made use of the parameters'characteristics and combining with non-monotone strategy, we put forward a new problem to solve large-scale algorithm.and each new method is the development of the old method.Finally,a new level vector is proposed and very effective.This article is divided into six chapters. In the first chapter, we introduce the basic idea of the trust region algorithm, the present situation of trust region on quadratic model and the conic model. Chapter II introduces the theoretical basis of conic model trust region algorithm and the new conic model trust region algorithm for solving the sub-problem.And we prove some of conic function theorem and list several testing classical optimization problem. In chapter III, by means of the dogleg algorithm for solving the sub-problem, a non-monotonic conic model of the new conic trust region algorithm is advanced, which is the development of the second model trust region algorithm and traditional conic model of trust region algorithm. Compared with them, the new conic model has been relatively good numerical results. And the global convergence and superlinear convergence are proved. In chapter IV, we give a new adaptive adjustment of trust region radius, and a new adaptive trust region method was proposed,which make up the deficiencies of the nonmonotone conic model trust region algorithm. And the global convergence algorithm is proved. Chapter V proposes a hybrid algorithm of the new conic-model trust region algorithm. With the new method, we can test some large-scale functions.Because of its ideal mumerical results, it is expected to solve large-scale optimization problems.The global convergence of the new algorithm is proved, and finally we analyze the numerical results of the algorithm.Because the approximation of the conic model is also a direction for improving and taking into account the importance of the level vector of parameters selection, finally, by means of approximation equivalence of the conditions of quasi-Newton between the quadratic model and the conic model, a new level vector is proved in chapter VI.