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Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. (English) Zbl 0872.90087
Summary: The paper presents concrete realizations of quasi-Newton methods for solving several standard problems including complementarity problems, special variational inequality problems, and the Karush–Kuhn–Tucker (KKT) system of nonlinear programming. A new approximation idea is introduced in this paper. The Q-superlinear convergence of the Newton method and the quasi-Newton method are established under suitable assumptions, in which the existence of ${F}^{\text{'}}\left({x}^{*}\right)$ is not assumed. The new algorithms only need to solve a linear equation in each step. For complementarity problems, the QR factorization on the quasi-Newton method is discussed.
##### MSC:
 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)