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A nonsmooth version of Newton’s method. (English) Zbl 0780.90090
Summary: Newton’s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton’s method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian for ${C}^{2}$-nonlinear programming is semismooth. Thus, the extended Newton’s method can be used in the augmented Lagrangian method for solving nonlinear programs.

##### MSC:
 90C30 Nonlinear programming 49J52 Nonsmooth analysis (other weak concepts of optimality) 49M15 Newton-type methods in calculus of variations 90-08 Computational methods (optimization)