Convergence Analysis Of Some Algorithms For Generalized Equation

In this work, we aim to find the solution of the generalized equation. For nonsmooth type generalized equation, we propose some exact and inexact algorithms and analyze the convergence of these algorithms under certain conditions; for underdetermined general-ized equation, we put forward the generalized Gauss-Newton’s method and analyze it’s convergence. The main work is organized in two chapters.In Chapter 2, in spirt of Uko’s work in [64] and QI’s work in [50], we propose some algorithms for solving generalized equation. These algorithms are made by generalized Ja-cobian instead of Frechet derivative. Under the condition that the function is nonsmooth, we give different conditions to get the convergence analysis of these algorithms, including linear convergence, superlinear convergence, quadratic convergence and 1+p-order conver-gence. Furthermore, we prove the local convergence of these algorithms and the existence and uniqueness of solution. At last, we take variational inequality problem as example for the numerical experiment, and the results show the effectiveness and convergence of the exact algorithm.In Chapter 3, we consider solving the generalized equation with surjective derivatives. Since there are more than one solution of generalized equation in every iteration for this situation, we propose generalized Gauss-Newton’s method. Considering the large-scale problem, it is difficult to find the solution with the least norm while doing the exact com-putation. Thus, here we only consider the assumption that the matrix of the derivative of function is of full row rank. Since the generalized equation is smooth and the space is finite dimensional space, we ensure the existness of the solution with least norm. At the same time, this shows that the algorithm is well-defined. By constructing a majorizing function, we set up the Kantorovich-type theorem under classical Lipschitz condition. Besides, the local convergence of this algorithm is also proved. Furthermore, under the assumption that the function satisfies Lipschitz condition with L average, we prove the semi-local and local convergence of the generalized Gauss-Newton’s method. At last, as applications, we ap-ply our results to some special cases, such as:we get Kantorovich-type theorem under the classical Lipschitz condition; convergence results under the γ-condition and Smale’s point estimate theory.