A Descent Algorithm For Minimization Problems And Nonlinear System Of Equations

In this thesis, we propose a descent algorithm for solving large-scale unconstrained mini-mization, and extend it to solve nonlinear system of equations. Under some mild condi-tions, we establish its global convergence. Numerical experiments are also reported, which illustrate that the proposed method is practical and promising.In the first chapter, we give the definition and some primary properties of the descent direction. We show that descent direction plays an important role in Newton-type method and conjugate gradient method. Moreover, some recent progress of the derivative-free scheme method for solving nonlinear system of equations are presented. To end this chapter, some important notation and symbols which used in the context are also included.In chapter 2, based on the sufficient descent PSB method of An, Li and Xiao, we develop a sufficient algorithm for unconstrained minimization. At each iteration, the generated directions is a combination of the gradient at current and previous steps. The attractive property is that the generated directions are always descent, and the property is independent of any line search rule. Under some appropriate conditions, we show that the method converges globally. We test the algorithm by using some problems from CUTEr library, which show that the proposed method is effective.In chapter 3, we extend the method which given in the previous chapter to solve nonlinear system of equations. At each iteration, the Jocobian information is not required, which saves the PC memory and improves the performance. Under some conditions, we establish the global convergence. Finally, we do some numerical experiments by using 44 problems, the numerical comparisons illustrated that the proposed method is competitive with the sate-of-the-art method DF_SANE.