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https://doi.org/10.1016/S0167-6377(01)00073-6
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. https://doi.org/10.1016/S0167-6377(01)00073-6 | 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 | URI: | http://scholarbank.nus.edu.sg/handle/10635/102647 | ISSN: | 01676377 | DOI: | 10.1016/S0167-6377(01)00073-6 |
Appears in Collections: | Staff Publications |
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