On locally-Lipschitz quasi-differentiate functions in Banach-spaces

Pallaschke, Diethard; Recht, Peter; Urbański, Ryszard

doi:10.1080/02331938608843131

Cited by 30 publications

(16 citation statements)

References 2 publications

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“…Roughly speaking, it is based on the same philosophy of a ''differential calculus without linearity'', that is allowing a larger set of firstorder approximants than that of linear ones, but it involves a different approximation concept, relying on the use of Fre´chet bornology in the defining limit constructions. For this possibility, which according to results obtained in [1] becomes interesting only for non-locally Lipschitz functions as far as working in finite-dimensional spaces, a systematic analysis of advantages is still lacking, up to the author's…”

Section: Introductionmentioning

confidence: 97%

“…The purpose of the study exposed in the present article is essentially to carry on investigations on a differentiation notion that were undertaken in [1], in the light of certain recent achievements in variational analysis. Such differentiation notion is a natural strong variant of the well-known one of quasidifferentiability, this latter having been studied and employed in many researches collected in a good deal of works during the last two decades.…”

Section: Introductionmentioning

confidence: 99%

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Fréchet quasidifferential calculus with applications to metric regularity of continuous maps

Uderzo¹

2005

Optimization

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To cite this article: Amos Uderzo (2005) Fréchet quasidifferential calculus with applications to metric regularity of continuous maps, Optimization, 54:4-5, 469-493, A strong variant for the notion of quasidifferentiability of functions is considered in the light of recent achievements in variational analysis. Several characterization results, some calculus rules and examples are provided. Then a generic result for the corresponding subdifferentiability notion is established in general Banach spaces. Subsequently, such notion is employed in the study of generalized differential criteria for metric regularity. Necessary and sufficient conditions, along with a precise estimation of the norm of metric regularity, are obtained in the case of continuous maps between Banach spaces via a strong slope approach. All these results do not require Asplundity assumptions.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Fréchet quasidifferential calculus with applications to metric regularity of continuous maps

Uderzo¹

2005

Optimization

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“…13, although Y = R in Ref. 13; therefore, its proof is omitted. The compactness of αB implies that there exists β ∈ (0, ηα/6Mk 1 ) such that f (x 0 + x, u) − f (x 0 + x, u 0 ) ≤ (η/3M) α, for all x ∈ αB, u ∈ B U (u 0 , β).…”

Section: If This Claim Is Not True Thenmentioning

confidence: 99%

“…For the proof, we need the following proposition which is indicated by Theorem 2.1 in Ref. 13, although Y = R in Ref. 13; therefore, its proof is omitted.…”

Section: If This Claim Is Not True Thenmentioning

confidence: 99%

Implicit Function Theorems for Nondifferentiable Mappings

Bian

2006

J Optim Theory Appl

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“…(1.1) f'(x;d) = p (d)-p_(d) = max <v,d> -max <w,d>, ve3f (x) we-af(x) where both of p^d) and p 2 (d) are sublinear operators, i.e., as the sum form of a pair of sublinear operator and superlinear operator, or as the difference form of two sublinear operators, [4], [8], [13]. This kind of structure of derivatives of quasidifferentiable function brings on that a quasidifferential of a quasidifferentiable function, called bidifferential also in [7], is not unique, but the quasidifferential class of equivalence of a quasidifferentiable function is unique.…”

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confidence: 99%