Spectral-scaling Quasi-newton Method For Solving Nonlinear Monotone Equations

The quasi-Newton method is one of the most efficient method for solving unconstrained optimization problems and nonlinear equations because it is of favorable numerical experience results and fast theoretical convergence. However, it is well-known that the matrices produced by the quasi-Newton method are dense and tend to ill-condition when the method is used to solve large-scale problems. To overcome this drawback, this thesis employs the spectral-scaling strategy to reduce the condition numbers and to prevent the quasi-Newton matrices from ill-condition. Based on the above consideration, by employing the quasi-Newton update and the limited memory technique, the thesis proposes four practical spectral-scaling BFGS algorithms and establishes their global convergence results under mild conditions. Moreover, the theoretical analysis shows that the BFGS algorithms using spectral-scaling technique can practically reduce the condition-numbers of the quasi-Newton matrices.In Chapter 2, by combining the projected idea of super plane and modifying BFGS algorithm, this thesis proposes a spectral-scaling modifying BFGS algorithm for solving monotone nonlinear equations and establishes the global convergence for the proposed algorithm. And some numerical results are reported for comparing the performance of the proposed algorithm with the MBFGS algorithm. In Chapter 3, by using the cautious BFGS updating technique, this thesis proposes a spectral-scaling cautious BFGS algorithm, and then obtains the global convergence results and reports some numerical experiments. In Chapter 4, to solve large-scale nonlinear equations, this thesis proposes two spectral-scaling BFGS algorithms with limited memory strategy for monotone nonlinear equations by employing the spectral-scaling technique and the limited memory technique, and then establishes their global convergence and reports some numerical experiments. The numerical results show that the proposed algorithms can effectively solve large-scale nonlinear equations.