A Proximal Bundle Method For Nonsmooth And Nonconvex Constrained Optimization

Nonsmooth optimization (NSO) problems are in general difficult to solve，even when they are unconstrained. Among algorithms for NSO, we mention thesubgradient, cutting-planes, analytic center cutting-planes (ACCP) and bundlemethods.Bundle and ACCP methods are stabilized versions of the cutting-planesmethod, and they are currently recognized as the most reliable NSO algorithms.The Bundle methods are at the moment considered as one of the most efficientmethods for solving nonsmooth optimization problems.Global convergence inconstrained optimization algorithms has traditionally been enforced by the useof parameterized penalty functions. Recently, the filter strategy has beenintroduced as an alternative. At least part of the motivation for filter methodsconsists in avoiding the need for estimating a suitable penalty parameter, whichis often a delicate task. In this paper, we demonstrate that the use of aparametrized penalty function in nonsmooth convex optimization can be avoidedwithout using the relatively complex filter methods. We propose an approach whichappears to be more direct and easier to implement,in the sense that it is closerin spirit and structure to be the well-developed unconstrained bundle methods.This article is divided into three parts. The first part are the priorknowledge, in this section,mainly to brief the relevant knowledge on the bundlemethod, include the sugradient method and cutting planes method.Pave the wayfor the introduction of the bundle method.The second part brief a proximal bundlemethod for nonsmooth convex unconstrained optimization. An infeasible bundlemethod for solving nonsmooth convex constrained optimization problem is alsopresented. In the following, the problem changes into the unconstrainedminimization of the improved function. But with the generation of new serioussteps, the improved function must be defined again. In this method, even theserious points generated in iteration process and the first original point areinfeasible, the convergence of the algorithm also is ensured. The last partgeneralize the proximal bundle method of nonconvex constrained optimization, and the algorithm be presented. The convergence analysis of the algorithm isalso given.