Study On Levenberg-Marquardt-type Algorithms For Solving Smooth And Nonsmooth Equations

The Levenberg-Marquardt(LM)algorithm is a classical and popular approach for solving ill-conditioned nonlinear equations.Since the 40s of the last century,a lot of important research results have been obtained on LM algorithm.But so far,most of the references on LM algorithm consider the smooth equations.Therefore,it is of signicant importance for us to study the LM algorithm for nonsmooth equations.In this thesis,we propose Levenberg-Marquardt-type algorithms for smooth and nonsmooth equations separately,and we analyze the convergence of the algorithms.The main results of this dissertation are summarized as follows:1.Chapter 1,mainly gives the introduction and current research of LM algorithm,including the basic ideas and research progress.In the end of Chapter 1,the main research work of this paper is summarized.2.Chapter 2 first describes a weaker condition.Then we recall the convergence of trust region algorithm and review some definitions and results in nonsmooth analysis.3.The contents of Chapter 3 are mainly about smooth equations.Based on the trust region technique,we propose a parameter-adjusting LM algorithm,in which the LM parameter is self-adjusted at each iteration based on the ratio between the actual redu-ction and the predicted reduction.Under the level-bounded condition,we prove the global convergence.We also propose a modified parameter-adjusting LM algorithm by altering the descent direction.4.In chapter 4,based on the results of nonsmooth Newton method,we construct a nonsmooth parameter-self-adjusting Levenberg-Marquardt algorithm and discuss its global convergence.Under the BD-regular condition,we prove that algorithm is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations.5.Finally,numerical results show that parameter-self-adjusting Levenberg-Marquardt algorithms are very efficient for smooth equations.Some preliminary numerical results for solving nonlinear complementarity problems are presented.We compare and analyze the numerical results.