Set-Valued Analysis

Aubin, Jean‐Pierre; Frankowska, Hélène

doi:10.1007/978-0-8176-4848-0

Cited by 1,124 publications

(1,639 citation statements)

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“…defined via the contingent cone (2.1) to the graph of F at the point (x,ȳ); see [3,35] for various properties, equivalent representation, and applications. The coderivative of F at (x,ȳ) ∈ gph F is introduced in [22] as a mapping D * F (x,ȳ) : IR m → → IR n with the values…”

Section: Tools Of Variational Analysismentioning

confidence: 99%

Generalized Newton’s method based on graphical derivatives

Hoheisel

Kanzow

Mordukhovich

et al. 2012

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B-differentiable versions of Newton's method for nonsmooth Lipschitzian equations.

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Section: Tools Of Variational Analysismentioning

confidence: 99%

Generalized Newton’s method based on graphical derivatives

Hoheisel

Kanzow

Mordukhovich

et al. 2012

Nonlinear Analysis: Theory, Methods & Applications

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show abstract

“…Then the closedness of C implies that ξ is a mapping from Ω to C. Since C is a subset of a separable Banach space X, if T is a continuous random operator then, by [1,Lemma 8.2.3], the mapping ω → T (ω, f (ω)) is a measurable function for any measurable function f from Ω to C. Thus {ξ n } is a sequence of measurable functions. Hence ξ : Ω → C, being the limit of the sequence of measurable functions, is also measurable [3,Remark 2.3].…”

Section: Preliminariesmentioning

confidence: 99%

Implicit random iteration process with errors for asymptotically quasi-nonexpansive in the intermediate sense random operators

Saluja¹

2012

OpMath

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Abstract. In this paper, we give a necessary and sufficient condition for the strong convergence of an implicit random iteration process with errors to a common fixed point for a finite family of asymptotically quasi-nonexpansive in the intermediate sense random operators and also prove some strong convergence theorems using condition (C) and the semi-compact condition for said iteration scheme and operators. The results presented in this paper extend and improve the recent ones obtained by S. Plubtieng, P. Kumam and R. Wangkeeree, and also by the author.Keywords: asymptotically quasi-nonexpansive in the intermediate sense random operator, implicit random iteration process with errors, common random fixed point, strong convergence, separable uniformly convex Banach space.Mathematics Subject Classification: 47H10, 47J25.

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