On Convergence Analysis Of Inexact Smoothing Levenberg-marquardt Algorithm For Non-smooth Constraint Equations

This paper proposed an inexact Levenberg-Marquardt algorithm for solving non-smoothconstraint equations and studied its convergence.Nonsmooth constraint equations have many widely applications. For example, nonlinearcomplementarity, variational inequalities and semi-infinite programming, can be transformed intothis problem. The problem of solving non-smooth constraint equations is a very important researchtopic in the field of optimization. This problem occurs in a large number of engineering andscientific experiments, and have a very wide range of applications in many areas. In addition, it isclosely linked with many optimization problems. Conventional (inexact) Levenberg-Marquardtalgorithm is one of classical and effective methods for solving smooth (i.e., continuouslydifferentiable) equations, and is mainly used to solve the system of equations without constraints.However, in many practical situations, the related functions (including the functions determing theconstraint set) in the considered system do not have smooth nature, there exist the error in thesolving large-scale problems, and the implementation of algorithm also spend a large amount oftime, hence some existing algorithms such as smoothing exact Levenberg-Marquardt algorithmsand traditional inexact (smooth) Levenberg-Marquardt algorithms can not be used directly. Thus,we need to propose inexact smoothing Levenberg-Marquardt algorithms to solve the system ofconstrained nonsmooth equations.In this paper, we first review the evolution of the Levenberg-Marquardt algorithms, andanalyzes the advantages and disadvantages of the existing algorithms. The problem of solvingnonsmooth equations with constraints are proposed. To this end, we transform the original probleminto an equivalent unconstrainted system via some smoothing technique. Based on this, we presentan inexact Levenberg-Marquardt algorithm. First of all, we give inexact Levenberg-Marquardtalgorithm with unit step. We show that this algorithm enjoys superlinear convergence or quadraticconvergence properties under local error bound conditions. Further, in order to obtain the globalconvergence of the algorithm, an inexact Levenberg-Marquardt algorithm with Armijo’s stepsizerule is presented. We prove that the presented algorithm also has local superlinear or quadriticconvergence properties under local error bound conditions.