Research On Some Nonlinear Conjugate Gradient Methods

Conjugate gradient method, which is an efficient way for unconstrained optimization problems, is in common use in optimization. It is of many advantages, such as simplifying algorithm, low requirement of storage, etc. It is qualified to do large scale of optimization and widely used in the fields of oil exploration, atmospheric modeling and aerospace, etc.Chapter 1 explains several ways of solving the problem of unconstrained optimization which are commonly used in optimization theory. Further more, chapter 1 also presents brief information about conjugate gradient method.Chapter 2 discusses the WYL conjugate gradient method. Under proper conditions, the global convergence of the WYL formula with the ATLS line search, modified strong wolfe-powell line search, modified Armijo line search and modified Armijo-Goldstein line search are proved, which improves the theory of WYL method.Chapter 3 and 4 put forward two modified PRP formulas for parameter. Then on the basis of these two formulas, proposes two modified PRP conjugate gradient methods, since the convergence of the PRP conjugate gradient methodsis not satisfactory. These two methods can automatically ensure that the formula for parameter is non-negative and of certain advanced quality. Global convergence results for the two proposed formulas with some inexact line searches. The preliminary numerical experiment also shows the effectiveness of the new algorithms.On the basis of the research in Chapter 3 and combining LS conjugate gradient method, Chapter 5 proposes a modified LS conjugate gradient method by using a new modified LS formulas for parameter for unconstrained optimization problems. The global convergence of the proposed formula with the strong Wolfe-Powell line search and the generalized Wolfe-Powell line search are discussed. Numerical results show that the method is promising.