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Journal of Convex Analysis 25 (2018), No. 4, 1075--1104
Copyright Heldermann Verlag 2018

Quasi-Newton Methods for Solving Nonsmooth Equations: Generalized Dennis-Moré Theorem and Broyden's Update

Samir Adly
Laboratoire Xlim, Université de Limoges, 123 Avenue Albert Thomas, 87060 Limoges, France

Huynh Van Ngai
Dept. of Mathematics, University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam

We study the quasi-Newton method by using set-valued approximations for solving generalized equations without smoothness assumptions. The set-valued approximations appear naturally when dealing with nonsmooth problems, or even in smooth cases, data in almost concrete applications are not exact. We present a generalization of the classical Dennis-Moré theorem, which gives a characterization of the q-superlinear convergence of the quasi-Newton iterates. The local linear and superlinear convergences of the method, especially, a modification of the Broyden update method are investigated. We present an example showing that the classical Broyden update method is no longer linearly convergent when the function involved in the nonlinear equation is not smooth. A modified version of the Broyden update is proposed and its convergence is proved. These results are new, and can be considered as both an improvement and an extension of some results appeared recently in the literature on this subject.

Keywords: Quasi-Newton methods, Broyden's update, Dennis-Moré theorem, metric regularity, generalized equations.

MSC: 47N10, 49J40, 49J53, 65K10, 65K15, 65J15, 90C30

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