Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

• 成果类型: Article
• 署名作者: Meng, K. W.; Li, M. H.; Yao, W. F.; Yang, X. Q.
• 署名单位: Southwestern University of Finance & Economics - China; Chongqing University of Arts & Sciences; Hong Kong Polytechnic University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01568-0
• 发表日期: 2021
• 页码: 455-489
• 关键词: directional metric regularity; optimality conditions; STABILITY; coderivatives; systems; MODULI; map

摘要

In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.