Lipschitz-like property relative to a set and the generalized Mordukhovich criterion
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Lipschitz-like property relative to a set and the generalized Mordukhovich criterion K. W. Meng1 · M. H. Li2 · W. F. Yao3 · X. Q. Yang3 Received: 29 November 2019 / Accepted: 16 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020
Abstract 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.
This paper is dedicated to Professor Marco A. López on the occasion of his 70th birthday.
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X. Q. Yang [email protected] K. W. Meng [email protected] M. H. Li [email protected] W. F. Yao [email protected]
1
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
2
School of Mathematics and Big Data, Chongqing University of Arts and Sciences, Yongchuan, Chongqing 402160, China
3
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
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K. W. Meng et al.
Keywords Lipschitz-like property relative to a set · Projectional coderivative · Mordukhovich criterion · Level-set mapping · Graphical modulus Mathematics Subject Classification 49J53 · 90C30 · 90C31
1 Introduction Stability theory of set-valued mappings has been extensively investigated. The monographs [5,9,24–26] contain a comprehensive presentation for the derivation of conditions ensuring various stability properties of set-valued mappings, including the Lipschitz-like property. The Lipschitz-like property of set-valued mappings (known also as Aubin property, or pseudo-Lipschitz property) has been introduced in [3]. The Lipschitz-like property has been well studied in the literature by virtue of Mordukhovich criterion, which was initially developed by [22] and a more direct proof was given in [26] by using the basic variational analysis tools. One important assumption when Mordukhovich criterion is applied is that the candidate parameter under consideration is in the interior of the domain of set-valued mappings. It is worth
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