## Newton methods for solving nonsmooth equations via a new subdifferential.(English)Zbl 1031.90069

Summary: A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods and inexact-Newton methods for solving the system of nonsmooth equations and for solving the system of equations of smooth compositions of nonsmooth functions, are developed. The $$Q$$-superlinear convergence of Newton methods and the $$Q$$-linear convergence of inexact-Newton methods are shown. The present Newton methods and inexact-Newton methods could be viewed as the extensions of previous ones with same convergent results.

### MSC:

 90C53 Methods of quasi-Newton type 49J52 Nonsmooth analysis 65H10 Numerical computation of solutions to systems of equations

### Keywords:

Newton methods; nonsmooth equations; subdifferential
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