Mathematical Programming

, Volume 104, Issue 2–3, pp 635–668 | Cite as

Subgradient of distance functions with applications to Lipschitzian stability

Article

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.

Keywords

Variational analysis and optimization Distance functions Generalized differentiation Lipschitzian stability 

Mathematics Subject Classification (1991)

20E28 20G40 20C20 

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA

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