Subgradient of distance functions with applications to Lipschitzian stability |
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Authors: | Boris S Mordukhovich Nguyen Mau Nam |
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Institution: | (1) Department of Mathematics, Wayne State University, Detroit, MI 48202, USA |
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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.
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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Keywords: | Variational analysis and optimization Distance functions Generalized differentiation Lipschitzian stability |
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