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Subgradient of distance functions with applications to Lipschitzian stability

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Abstract

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich & Nguyen Mau Nam

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  1. Boris S. Mordukhovich
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  2. Nguyen Mau Nam
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Correspondence to Boris S. Mordukhovich.

Additional information

Dedicated to Terry Rockafellar in honor of his 70th birthday.

This research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451158.

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Mordukhovich, B., Nam, N. Subgradient of distance functions with applications to Lipschitzian stability. Math. Program. 104, 635–668 (2005). https://doi.org/10.1007/s10107-005-0632-1

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  • DOI: https://doi.org/10.1007/s10107-005-0632-1

Keywords

Mathematics Subject Classification (1991)