The Study Of Newton’s Method For Two Kinds Of Problems

This thesis will study Newton methods for a class of problems of solving nonlinear equations and a class of bilinear constrained optimization problems. The main works in this thesis are listed as follows.Firstly, a new modified Newton method is presented, in which the iteration direction is modified by introducing a controlling parameter, in order to overcome the ill-conditioning problem. This ill-conditioning problem is that the initial solution is near the true solution but the final iteration solution is away from the true solution when solving nonlinear equations by the traditional modified Newton method.Furthermore, the convergence of the proposed modified Newton algorithm and the error estimates in complete normed linear space are established. At last, the numerical experiments are reported to show that the new modified Newton algorithm is effective.Secondly, a semi-smooth Newton method for a class of bilinear constrained optimization problems is proposed. The concrete forms of the first and second order optimality conditions for the special class of optimization problems are presented,where the strict complementarity relaxation condition is not considered. The class of optimization problems is reformulated as a system of nonlinear equations with original variables and the Lagrange multipliers based on the associated quadratic augmented Lagrange function and a semi-smooth Newton algorithm is constructed.The quadratic convergence of the algorithm is proved based on semi-smooth analysis theory and the uniqueness theorem of solutions of matrix. Finally, the numerical results for several bilinear constrained optimization problems are reported.