Research On The Shamanskii Modification Of The Newton Method

Three parts are included in this dissertation. We split three parts into four chapters.The first part includes two chapters. Chapter one is the introduction of this dissertation, which introduces the Shamanskii modification of the Newton method and points out that this method we study in this dissertation is the following iterative scheme:whereThe matrix H(xip) denotes the Hessian or a suitable modification of it, which is recomputed every p iterations, and the step k along the search direction dk = -Dkg(xk) is computed by means of some monotonical line search. Techniques for ensuring the global convergence of the Shamanskii modification of the Newton method had not yet been studied since this iterative scheme was proposed. But recently, F.lampariello and M.sciandrone in Ref[10] proposed an Armijo - type line search as follows:As far as we know, it was the first time that the global convergence of the Shamanskii modification of the Newton method with the Armijo - type linesearch was proved by them. Here, we generalize the global convergence result in Chapter two.The second part is Chapter three. In this chapter, we point out that the Armijo - type line search will make the step k very small even close to zero in some cases, which leads to the pathological problem. For this reason, we present a new Wolfe - type line search as follows:The Wolfe - type line search can validly avoid k approaching zero. Furthermore, the global and superlinear convergence of the Shamanskii modification of the Newton method with the new line search are proved under the weaker conditions than those in Ref[10](i.e., we avoid the hypotheses on the forcing function). Moreover, our convergence result is more general than that in Ref[10].The third part is Chapter four. In this chapter, we generalize the Shamanskii modification of the Newton method. We substitute pt for the positive integer p, where pt satisfiesThus, the need for the Hessian matrix is reduced significantly. The general Shamanskii modification of the Newton method is defined by the iterationandThe global and superlinear convergence of the general Shamanskii modification of the Newton method are proved in this dissertation.