Subgradient of distance functions with applications to Lipschitzian stability

• 成果类型: Article
• 署名作者: Mordukhovich, BS; Nam, NM
• 署名单位: Wayne State University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-005-0632-1
• 发表日期: 2005
• 页码: 635-668
• 关键词: metric regularity; PRINCIPLE; subdifferentials; MULTIFUNCTIONS; optimization; SPACES; sets

摘要

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 Frechet-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.