On Global Convergence Of Nonlinear Conjugate Gradient Method

Three parts are included in this dissertation, we split three-parts into four chapters.The first part includes chapter one and chapter two. In chapter one we can see the introduction of nonlinear conjugate gradient method's research content, research valuation, research circumstance and the main work of this paper. In chapter two, global convergence properties of three-term conjugate gradient method with new-type line search of unconstrained optimization is discussed.Two assumptions are given as follows:(a) f(x) is bounded below at the level set L = {x € Rn\f(x) < f(x0)}. (6) There exists a neighbourhood U containing the level set L , such that / is continuously differentiable and its gradient is Lipschitz continuous on U , that is , there exists L > 0 such thatThe new-type line search:where 0 < ρ < σ < 1, and Armijo - type line search are also introduced. Using the restar technique of Dai and Yuan together with the above line searches , we could get the global convergence of three-term conjugate gradient method under the above two assumptions. This method includes the classical(two-term) conjugate gradient method and Beale - Powell restart algorithm as its special forms. This work also establish theoretical ground of rapid three-term conjugate gradient algorithm.Part two is chapter three. In this chapter , the global convergence result of a class of conjugate gradient methods is discussed. In this class of conjugategradient methods, βk satisfieswhereStep-size αk is obtained by Grrippo-Lucidi line search. The global convergence of this class of conjugate gradient methods is obtained under the above conditions. The numerical results of this algorithm show that this method retain the same good features of the Polak-Ribiere method.Part three is chapter four. In this chapter, the global convergence result for nonmonotone conjugate gradient methods is given. Firstly, we give the following nonmonotone line search:wherem(k) = min{m(k-1) + 1, M0}, m(0) = 0, M0 is a nonnegative integer,Under the above line search, we get the global convergence result of non-monotone conjugate gradient methods with βk satisfing:...