Trust Region Bundle Methods For Nonsmooth Optimization And Its Convergence Analysis

Currently the bundle method is more widely used by researchers for solving nonsmooth optimization problems.This method possesses the stability property to some extent,at the same time it can ensure the descent of the objective function.It is characterized by the ability to use the information in the bundle to retain the iteration information which have been obtained,which can prevent the loss of the "best" point and make it easy to find the optimal solution to the problem.In this paper,we mainly study the trust region bundle method-a kind of the bundle methods.This method constructs a new subproblem by using trust region idea.Under the Salter constraint qualification condition,the constraint problem is transformed into unconstrained optimization problem.By using the dual space idea,we transform the primal problem and the dual problem to each other,and then obtain their optimal solutions respectively.Furthermore,we obtain the important derivative conclusions by studying the expression of solution to the subproblem.Finally,we propose the trust region algorithm and analyze the convergence of the algorithm.By employing the trust region idea,this article is mainly based on the construction of iterative subproblem.From the perspective of dual space,based on the results of unconstrained optimization,we transform the problem of nonsmooth optimization with constraints into an unconstrained optimization problem in order to find the solution.The paper is divided into four parts.The main contents are as follows:In the first chapter,in order to facilitate the understanding of the article,we first give some basic concepts and conclusions which have been reached.Then,we introduce some basic methods for solving nonsmooth optimization problems.For example,the steepest-descent method,the black-box method,the subgradient methods,the cutting-plane method and the summary of general bundle methods.These methods are the theoretical basis for our further research.In the second chapter,with the help of the existing stability principle and the obtained function information,we construct the subproblem by imitating the idea of trust region.When the elements in the bundle of information are too much,we use the aggregation technique to deal with the model.In the third chapter,we study the Lagrangian function of trust region subproblem from the perspective of dual space.In the dual space,we discuss the relation between the primal problem and the dual problem in order to obtain the related expression of the optimal solution of the subproblem.In addition,we give three important derivative conclusions.In the fourth chapter,the specific trust region bundle algorithm is given in this section.We proceed under two situations,the first condition is that the trust region bundle algorithm generates an infinite number of descent steps,the second condition is that the trust region bundle algorithm generates an infinite number of zero steps immediately after the last descent iteration point.Under these two cases,we give the convergence results of the specific trust region bundle algorithm.