Several Numerical Optimization Methods For Nonlinear Equations

Nonlinear optimization is an important branch of operational research. Nonlinear problems come from many areas of science and engineering calculation, such as weather forecast problem, nonlinear finite element problem, nonlinear fracture and petroleum geological exploration problem, etc. Nonlinear equations and nonlinear optimization problems are closely related, so the study of the numerical solution of nonlinear equations based on the algorithms of nonlinear optimization problem has theoretical significance and practical value. Common algorithms for solving nonlinear optimization problems conclude Newton method, quasi-Newton method, conjugate gradient method, trust region method, etc. This thesis mainly forces on the limited memory BFGS algorithm, the inexact backtracking algorithm and projecting conjugate gradient algorithm. The main contents and achievements are shown as follows:(1) Firstly, we do some study about the limited memory BFGS algorithm for solving nonlinear equations, and propose a new limited memory BFGS algorithm, then analyze its decline property and global convergence property etc, finally design some numerical experiments. The numerical results show that the new algorithm does better in solving large scale nonlinear equations than the normal BFGS algorithm.(2) Firstly, we give a revised search direction, and propose a new algorithm with inexact backtracking line search technique, then prove the decline property and global convergence properties of the algorithm, and finally design some numerical experiments. The numerical results show that the performance of the new algorithm is better than the normal BFGS algorithm.(3) Firstly, we put forward a new projecting conjugate gradient algorithm, and then the sufficient decline property, the trust region property and the global convergence property of the algorithm are proved, and finally design some numerical experiments. Numerical results show that the new algorithm performs better than normally PRP algorithm in solving the large-scale nonlinear equations.