On Theory Of Projeced Gradient Methods For Convex Constrained Optimization Problems

Optimization method is an important part of operations research. It has wide application to many fields, such as natural science, social science, practical production, engineering design and modern management, etc. This paper is devoted to studying the theoretical properties of projected gradient methods for convex constrained optimization problems, including convergence properties of nonmonotone spectral projected (SPG) gradient methods, error bound on projected gradient methods and convergence properties of inexact hybrid projected gradient methods. It is composed of four chapters.Chapter 1 is the introduction of this paper, which introduces the the projected gradient method, the error bounds estimation and the main results obtained in this paper.Chapter 2 is committed to develop the convergence properties of nonmonotone spectral projected gradient methods. In a recent paper, a nonmonotone spectral projected gradient method was introduced by Birgin, Martinez and Rayclan (2000)for the minimization of differentiable functions on closed convex sets and extensive presented results showed that this method was very efficient. In this chapter, we give a more comprehensive theoretical analysis of the SPG method. In doing so, we remove various bouncledness conditions that are assumed in existing results, such as boundedness from below of f., boundedness of xk or existence of accumulation point of {xk}. If f(:) is uniformly continuous, we establish the convergence theory of this method andprove that the SPG method forces the sequence of projected gradients to zero. Moreover, we show under appropriate conditions that the SPG method has some encouraging convergence properties, such as the global convergence of the sequence of itetates generated by this method and the finite termination etc. Therefore, these results show that the RPG method is attractive in theory. In Chapter 3, some error bounds estimations of the projected gradient methods are obtained. We first consider the optimal value function defined by the projected gradient subproblem (QP(x))and study some properties of this value function. Under different conditions, such as f(.) is strongly monotone, V/(-) is monotone and V/(-) is pseudo-monotone, \ve show that the value function provide global or local error bounds for the distances from the iterative point to the solution set of the minimization problem. Using the error bounds, we give a condition for convergence of the sequence of iterates generated by projected gradient methods.In Chapter 4, we study the covergence properties of inexact hybrid projected gradient methods. It is well known that inexact variable metric method is a generalization of the nonmonotone spectral projected gradient method. In this chapter. we extend the convergence theory of the inexact variable metric method in Birgin, Martinez and Rayclan(2003). Without thp boundedness condition of the level set, a convergence theory is obtained under weaker assumption. that is, f(-) is uniformly continuous. Furthermore, combining inexact variable metric method with some technique of M.V. Solodov and B.F. Svaiter(1999). we give a new method and establish the global convergence ofthe sequence of itetates generated by this method under the condition f(.) is pseudo convex.