Two Nonmonotone Algorithms For Composite Nonsmooth Programming Problems

A class of nonsmooth composite minimization problems (P) min h(f(x)) are considered in this thesis, where each f : Rn - Rn is a locally Lipschitzian function, and h : Rn - R is a continuously differentiable convex function.In practice, many problems from engineering and statistics problems fall into the class of problems (P). For instance, the problem of solving the system of non-smooth equations and the least squares problem with nonsmooth data are special cases of (P). Because of its practical importance, this class of problems (P) has received a lot of attention from the research community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for (P) and also provided the convergence analysis to support their algorithmic proposals. Motivated partly by their work, we present two nonmonotone algorithms for this class of nonsmooth composite minimization problems which based on line search method and trust region method respectively in this thesis. After a brief introduction of the problems (P) in the first Chapter, a nonmonotone line search algorithm for this class of problems is proposed in the second Chapter, and the global convergence of the algorithm is also studied in this chapter. In the third and the final Chapter, we provide a nonmonotone trust region method as well as the global convergence analysis of the new algorithm.