A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems

Abstract

A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.