Abstract
We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ℝn,D being an open bounded subset of ℝn. 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.
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Jongen, H.T., Klatte, D. & Tammer, K. Implicit functions and sensitivity of stationary points. Mathematical Programming 49, 123–138 (1990). https://doi.org/10.1007/BF01588782
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DOI: https://doi.org/10.1007/BF01588782