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A Nonlinear Extension of Hoffman's Error Bounds for Linear Inequalities

C. Zlinescu

(cf. Prof. Christiane Tammer), Martin Luther Universität, Halle-Wittenberg, FB Mathematik und Informatik, Institut für Optimierung und Stochastik, 06099 Halle (Saale), Germany.

In a recent paper Li and Singer (1998) introduced the notion of global error bound for a convex multifunction at a point of its domain. They showed the existence of such a global error bound when the image of the multifunction at the respective point is bounded and conjectured a result for the case when the image is not bounded. In this paper we solve their conjecture with a positive answer. For this we establish a criterion for the existence of a global error bound using the Pompeiu-Hausdorff excess. We also improve slightly some results of Li and Singer and introduce a gage associated to a multifunction similar to that for well-conditioning of convex functions, with similar properties.

Key Words: Asymptotic cone; bounded selection; convex multifunction; global error bound
History: Received: April 3, 2001; revision received: February 11, 2002;

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