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Mathematical Programming

, Volume 49, Issue 1–3, pp 123–138 | Cite as

Implicit functions and sensitivity of stationary points

  • H. Th. Jongen
  • D. Klatte
  • K. Tammer
Article

Abstract

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ℝ n ,D being an open bounded subset of ℝ n . LetF belong toL(D) and suppose that\(\bar x\) solves the equationF(x) = 0. In case that the generalized Jacobian ofF at\(\bar x\) is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of\(\bar x\) Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.

Key words

Implicit function stationary point strong stability Lipschitz continuity generalized Jacobian mapping degree 

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Copyright information

© The Mathematical Programming Society, Inc. 1990

Authors and Affiliations

  • H. Th. Jongen
    • 1
    • 4
  • D. Klatte
    • 2
  • K. Tammer
    • 3
  1. 1.Lehrstuhl C für MathematikRWTH AachenAachenFR Germany
  2. 2.Pädagogische Hochschule Halle-KötchenSektion Mathematik und PhysikHalle (Saale)GDR
  3. 3.Technische Hochschule LeipzigSektion Mathematik und InformatikLeipzigGDR
  4. 4.University of HamburgFR Germany

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