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Title: A further result on an implicit function theorem for locally Lipschitz functions
Authors: Sun, D. 
Keywords: Higher order approximation
Implicit function theorem
Locally Lipschitz function
Issue Date: May-2001
Citation: Sun, D. (2001-05). A further result on an implicit function theorem for locally Lipschitz functions. Operations Research Letters 28 (4) : 193-198. ScholarBank@NUS Repository.
Abstract: Let H:Rm × Rn → Rn be a locally Lipschitz function in a neighborhood of (ȳ,x̄) and H(ȳ,x̄) = 0 for some ȳ ∈ Rm and x̄ ∈ Rn. The implicit function theorem in the sense of Clarke (Pacific J. Math. 64 (1976) 97; Optimization and Nonsmooth Analysis, Wiley, New York, 1983) says that if πx∂H(ȳ,x̄) is of maximal rank, then there exist a neighborhood Y of ȳ and a Lipschitz function G(·):Y → Rn such that G(ȳ) = x̄ and for every y in Y, H(y,G(y)) = 0. In this paper, we shall further show that if H has a superlinear (quadratic) approximate property at (ȳ,x̄), then G has a superlinear (quadratic) approximate property at ȳ. This result is useful in designing Newton's methods for nonsmooth equations. © 2001 Elsevier Science B.V.
Source Title: Operations Research Letters
ISSN: 01676377
DOI: 10.1016/S0167-6377(01)00073-6
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