The Theoretical Study On A Class Of Nonsmooth Optimization Of Level Bundle Method

The problem of nonsmooth optimization is that at least one of the objective function and constraint functions is not a continuous and differentiable programming problem,it is an important branch of optimization theory and method,because it has no continuous differentiable properties,traditional differential concepts and optimization theories can no longer be suitable for use,so the corresponding methods for solving this kind of problems have always been the focus of the optimization theory research.The basic methods for non-smooth optimization problems include: general descent method ? steepest descent method?subgradient methods?cutting plane method?black box method?bundle method and so on.In these methods,the bundle method is a method which combines the descent and the stability,the advantage is that it can retain the information of the last iteration and form a bundle of information,so we don't have the possibility of losing the "best" point,thus the optimal solution of the problem can be found quickly and accurately.In this paper,we study mainly the level bundle method,which is one of bundle methods.It uses the information in the black box to construct a piecewise affine model of the objective function for the original problem,the level set as the constraint is used to construct subproblem which produces the next iteration point,as the number of iterations increases,we use the compression mode to control the size of the subproblem.The Lagrangian function and its dual problem are studied,and the expression of the optimal solution of original subproblem is obtained,three important conclusions and the convergence analysis for level bundle method are presented.In the first chapter,to understand the contents of the article better,first of all,some basic concepts?methods and conclusions related to the level bundle method are expounded,for example,convex function?constraint qualification?algorithm steps,etc.In the second chapter,the original problem studied in this paper is given,a piecewise affine model for the objective function is constructed,further the specific form of subproblem is put forward,its Lagrangian function and its dual problem is analyzed thoroughly,the explicit representation of the optimal solution is obtained.In the third chapter,further analysis for the subproblem is given,some important conclusions are drawn that are closely related to the convergence of the algorithm.In the fourth chapter,according to the corresponding important conclusions obtained in Chapter 3,we analyze the convergence of the proposed algorithm.The analysis of theconvergence of the algorithm are carried out in two parts.For the one part,we analyze the convergence of the problem,determine its convergence.For the other part,we give the concrete convergence points,and show that it is the minimum point of the original function.