Projection Type PRP Methods For Solving Monotone Nonlinear Equations

In this paper we study numerical algorithms for solving large-scale and mono-tone nonlinear equations. The monotone nonlinear equations has many applications, for example, the monotone variational inequality can be equivalently converted into monotone equations. In the past decade, numerical methods for monotone nonlin-ear equations have attracted many attention. There are many efficient methods for such problems including Newton methods, quasi-Newton methods and residual type methods. Newton methods and quasi-Newton methods are not efficient for solving large-scale problems since they need compute Jacobian or store matrices. Existing residual type methods include spectral residual methods and conjugate residual type methods. Numerical results have showed that conjugate residual type methods are more efficient than spectral residual methods. Residual type methods are suitable for solving large-scale problems because they do not require computing or storing matri-ces. However, existing conjugate residual type methods are derivative-free algorithms which are mainly based on some modified PRP methods or modified HS methods.In this paper, based on the standard unmodified PRP nonlinear conjugate gradient methods and the idea of hyperplane projection in, we propose a new projection type PRP conjugate residual method for solving monotone equations and prove its global convergence and Q-linear convergence. The paper is organized as follows.In Chapter 1, we simply introduce the background and the main work of the paper.In Chapter 2, we present a projection type PRP method. We introduce a new line search technique to guarantee some descent property of the proposed method, which can determine the stepsize and the search direction simultaneously. We adopt the hyperplane projection technique in to ensure global convergence of the method and prove that the iterative sequence converges to a solution of the equations.In Chapter 3, we further discuss the convergence rate of the proposed method. To this end, based on the idea of , we modify the line search proposed in Chapter 2 by sufficiently using the property of the PRP formula and the monotone structure of the problem. We establish the global convergence and Q-linear convergence rate of the method with this modified line search even for nonsmooth equations. Moreover, we also do some numerical experiments, which show that the proposed method is more efficient than some existing methods.