Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations

TLDR: Convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions and a hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is presented.

Abstract: This paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and the generalized Jacobian-based Newton methods, both the iterates and the corresponding function values are locally, superlinearly convergent. Globally, a limiting point of the iterate sequence generated by the damped, directional derivative-based Newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one, provided that the system is strongly BD-regular and semismooth at this point. In this case, the convergence is superlinear. A general attraction theorem is presented, which can be applied to two algorithms proposed by Han, Pang and Rangaraj. A hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is also presented.