Study On Singular Nonlinear Systems Of Equations And Nonlinear Optimization

This thesis focuses on the numerical method for solving the singular nonlinear system of equations and singular nonlinear optimization problem, including the extended ABS projection algorithm for solving nonlinear system of equations and subspace transforming method and the tensor BFGS method for solving the unconstrained nonlinear programming problem. The thesis is outlined as follows:1. In the second chapter, we study the extended ABS method for solving the singular nonlinear system of equations. Firstly we assume that Rank(F'(x*)) =n-1 in the first section of this chapter and uses the properties of the null space Null(F'(x*)) = u and Null(F'(x*)T) = v to set up a full rank nonlinear system of equations which has the same solutions with the original system of equations F(x) = 0. Because the functions T(x) are concerned with the accurate vectors u, v of the null space, we adopt the sequence of subproblems Tk(x) = 0 iterative methods where the subprob-lems are determined by the current iterate point's information xk,uk,vk.We prove that the numerical method given in this dissertation is local quadratic convergence. In the second section, we extend the rank defeat assumption of the first section to Rank(F'(x*)) = n — s,(s << n). Based on the approximate matrix sequence Uk,Vk, we set up the iteration of the second part of the sequence subproblems Tk(x) — 0 by adopting the similar idea in the second section. Finally we prove that the given numerical methods are still local quadratic convergence.2. In the third chapter, we discuss the space transforming method of the singular nonlinear system of equations. The main idea of the chapter is to adopt the space linear transforming method to construct the nonsingular models with the same solutions and the same dimensions T(x) = 0, and decompose this problem into sequence of subproblems, Tk(x). In the first section, an iterative algorithm of simple singular point is constructed. This nonsingular model T(x) = 0 uses one definite nonzero vector c* of the null space Null(F'(x*)). But because the nonzero vector c* is unknown, we need to form an iterate sequence of subproblems Tk(x) = 0 in the process of iteration,each Tk(x = 0) still uses modified ABS projection method and proves that the given numerical algorithm is still local quadratic convergent. In the second section, when we generalize the rank defeat assumption to Rank(F'(x*)) = n — s, (1 < s<