BFGS Method For Nonlinear Monotone Equations With Convex Constraints

In this thesis, we study the BFGS and limited memory BFGS method for solvingmonotone nonlinear equations with convex constraints, and then we establish the conver-gence theorem of the proposed method whose e?ectiveness can be verified by numericalexperiments.In chapter 1, we introduce Newton methods for unconstrained optimization nonlinearequations problems.We also describe some recent progress of BFGS method and limitedmemory BFGS method. Moreover, some primary properties of convex functions andconvex set are some included.In chapter 2, we propose a BFGS method for solving nonlinear monotone equationswith convex constraints. An attractive feature of the proposed algorithm is that it is notnecessary to compute or store the Jocobian matrix. Moreover, the proposed method is notneed to solve a linear system to determine the search directions at each iteration. Hence,it is potentially to solve non-smooth equations. Under some appropriate conditions, weprove the global convergence of the algorithm. At last, numerical experiments which showthe e?ciency are also reported.In Chapter 3, we propose limited memory BFGS method for solving convex con-strained nonlinear monotone equations. Compared to the algorithm in previous chapter,the proposed algorithm is not necessary to store any matrix, which it can improve itsperformance and suit for solving large-scale problems. Finally, we establish the globallyconvergence of algorithm, and test its e?ectiveness by using some large-scale problems.In Chapter 4, we give a summary of this paper and some further research topics.