On The Algorithms And Their Convergence Analysis For Nonlinear Equations And Variational Inequalities

This thesis includes three chapters, which mainly discusses the iterative algorithms for solving nonlinear equations problems and nonlinear variational inequality problems.In Chapter 1, we consider the nonlinear monotone equations problem. For this problem, Solodov and Svaiter(1998) proposed an algorithm which possesses good global convergence property. This algorithm first computes a search direction by solving a Newton-like equation, then uses an Armijo-like line search to get the hyperplane which strictly separates the current iterate from the solution set of the problem. The next iterate is obtained by projecting the current iterate onto the hyperplane. In this chapter, we modify the algorithm from the following two aspects: (1) We project the current iterate onto the intersection of two half spaces, which guarantees the new iterate is closer to the solution set of the problem theoretically; (2) we modify the parameters in the original algorithm so that the restrictions on parameters are weakened in the convergence analysis, which makes the new algorithm more easy to be implemented in practice. Under the standard assumptions that the underlying mapping is monotone and the solution set of the problem is nonempty, we show the global convergence of the improved algorithm. Furthermore, by using an error bound condition, we establish linear convergence of the proposed algorithm. We also give a sufficient condition to guarantee that the stepsize of the line search has a positive below bound. At last, we give an application of the improved algorithm to solve the variationalinequality problem and give the preliminary numerical experiments which show that the improved algorithm has good behavior.In Chapter 2, we deal with the constrained system of nonlinear monotone equations problem. Inspired by the Newton-type method proposed in the first chapter for unconstrained system of monotone equations, we give a new projection algorithm for solving the constrained system of nonlinear monotone equations with the following main features: (1) It has global convergence property under standard assumptions; (2) At each step, it only needs to solve a system of linear equations approximately so that the computional expenditure is not very expensive; (3) We introduce a kind of long stepsize rule to guarantee faster convergence rate. The numerical test results show that the new algorithm is not only feasible, but also very effective.In Chapter 3, the variational inequality problem is considered, for which D. Han (2003) presented a generalized proximal point algorithm with uses the following procedure: First obtain a trial point by solving the proximal subproblem inexactly, and then the next iterate by a projection step. It is proven that the algorithm possesses global convergence under suitable assumptions. We give an improved generalized proximal point algorithm by modifying the projection region, which guarantees that the new iterate is closer to the solution set of the problem and hence convergence rate is enhanced theoretically. The improved algorithm also possesses the following features: (1) The sequence generated by the algorithm has an expansion property with repect to the initial point; (2) The algorithm provides a sufficient and necessary condition to identify the existence of the solution of the problem by the behavior of the generated sequence; (3) Under suitable assumptions, the algorithm has the global convergence; (4) If the solution set of the variational inequality problem is nonempty, then the limit point of the sequence generated by the improved algorithm is just the projection...