Two Methods For Solving Monotone Nonlinear Equations With Convex Constraints

In this thesis, we propose a modified Polak-Ribière-Polyak (MPRP) method and scaled conjugate gradient (SCALCG) method for solving monotone nonlinear equations with convex constraints. Under weaker conditons, we establish the convergence theorem of the proposed two methods. We also do some numerical experiments to test the performance of the proposed methods.In chapter 1, we simply review the MPRP and SCALCG methods for solving unconstrained optimization problems. MPRP is a very efficent conjugate gradient method. A good property of MPRP method is its lower storage and fast convergence rate. It is suitable for solving large scale problems. SCALCG method can be regarded as a combination of conjugate gradient method and Newton-type method. It is also suitable for solving large scale problems. We also briefly introduce the background of the monotone nonlinear equations with convex constraints.In chapter 2, we propose a MPRP method for solving monotone nonlinear equations with convex constraints. Under reasonable conditions, we show that the proposed method is globally convergent. Moreover, we do some numerical experiments to show that the proposed method is efficient and stable. Finally, we make some improvement to the method. Our numerical experiments illustrate that the improved method has better numerical performance.In chapter 3, we propose a SCALCG method for solving monotone nonlinear equations with convex constraints. Under the same conditions as those in chapter 2, we prove its global convergence. We also do some numerical experiments show that the proposed methods are efficient and promising. Finally, we do some improvement to the method. A lot of numerical experiments illustrate that compared to the SCALCG method, the improved method has better numerical performance.