An Approximate Bundle Method For A Class Of Nonsmooth Nonconvex Constrained Optimization

Bundle methods have been shown to be highly efficient in solving nonsmooth optimization problems.Due to the complexity of the practical problems,it will be of great practical research value to extend the relevant results of bundle methods.In this work,we mainly study an approximate bundle method for a class of nonsmooth nonconvex constrained optimization.We consider the problem that the available information about objective and constraint functions at any point are approximate function values and approximate subgradient values,whose errors with exact values are required to be bounded.Since the primal problem is constrained,firstly,we use the penalty function method to transform the problem into an unconstrained one,in which the penalty parameter will stop updating in some step by adding constraint qualification.Then,according to the special constructrue of the objective and constraint functions,we do some convexification for the constructed penalty function,and by employing the proximal mapping we establish the relationship between the convexification model and the primal one.Afterwards,based on the obtained inexact information,we build an approximate linearization model of the convexification penalty function,and obtain the next candidate point by solving a penalized subproblem,and from the dual point of view we discuss the solution expressions.In addition,we introduce the compression technology to get some significant relations.On the basis of the redistributed bundle method and the inexactness of the available information we provide the concrete algorithm to solve our problem: approximate redistributed bundle method.Finally,we prove the eventual stabilizations of the relevant parameters,and by analyzing the convergence of the algorithm,we obtain the approximate optimality results under some conditions.