A further result on an implicit function theorem for locally Lipschitz functions

Authors:
Defeng Sun
Affiliations:
Department of Mathematics, National University of Singapore Singapore 117543, Singapore
Venue:
Operations Research Letters
Year:
2001

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Abstract

Let H:R^mxR^n-R^n be a locally Lipschitz function in a neighborhood of (y@?,x@?) and H(y@?,x@?)=0 for some y@?@?R^m and x@?@?R^n. 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 @p"x@?H(y@?,x@?) is of maximal rank, then there exist a neighborhood Y of y@? and a Lipschitz function G(.):Y-R^n such that G(y@?)=x@? 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 (y@?,x@?), then G has a superlinear (quadratic) approximate property at y@?. This result is useful in designing Newton's methods for nonsmooth equations.