An Approximate Proximal Splitting Bundle Methods For Minimizing Nonsmooth Nonconvex Functions

The bundle method is a method of combining descent and stability.The uniqueness of bundle method is that it can use the bundle of information to retain the previously obtained iterative information so that people do not lose “the best”point,which helps to solve the problem of non-smooth optimization.It is considered as one of the perfect methods so far and widely used in many fields,such as economy,machinery,engineering,biology and optimal control aspects,and is also very efficient.Sometimes bundle method is considered to be the stable variant of cutting plane method and steepest descent method.To solve different problems,many scholars have improved and generalized the bundle method,such as solving nonconvex problems by employing the method of solving convex problems and turning constrained optimization problem into unconstrained optimization problem,etc.Therefore,there are many variants of bundle method,which brings a great breakthrough of applications for this method.This paper we focus on the problem of minimizing nonsmooth nonconvex unconstrained optimization problem by using the approximate proximal splitting bundle method.The problem we study is unconstrained.On the basis of previous results,we refer to the literature of the former people to extend the exact problem to the inexact problem.We add the constructed approximate information into the considered objective function,and by making disturbance for function to observe the variation of the approximate optimal solution and the true value of the function.For the error accuracy in this article,we demand it should be bounded.In the first chapter,in order to facilitate the understanding of the article,we give the preliminary knowledge about the bundle method,such as some simple concepts that can be used,general overview of the bundle method and some simple ideas of the penalty function methods.In the second chapter,the function behavior is divided into two parts,convex behavior and concave behavior.It means that the information bundle is divided into non-negative linearization error and negative linearization error.By using these two information bundles,we respectively construct the piecewise affine functions,then we construct the penalty function and the subproblems for searching the next point.In order to produce the next iteration point,we study the dual problem of the primal subproblem,and obtain the explicit expression of the optimal solution to the primal subproblem.In the third chapter,we give the concrete approximate proximal splitting bundle method,and describe briefly the algorithm.In the fourth chapter,we analyze the convergence of the proposed algorithm,which is divided into two parts,one focuses on the optimal solution to the subproblem and the other aims at the finite convergence analysis of the algorithm.It is proved that the algorithm can stop at a certain point if the proximal stable condition is satisfied,and the approximate optimality result can be obtained under some conditions.