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

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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.

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Acknowledgements

The authors are grateful to the guest editor and the anonymous reviewers for their valuable comments and suggestions toward the improvement of this paper. The authors would like to thank Professor Rockafellar for posing to us a few research questions on the Lipschitz-like property relative to a set during a conference held in Beijing in 2008, most of which have been addressed in this paper. Kaiwen Meng’s work was supported in part by the National Natural Science Foundation of China (11671329) and by the Fundamental Research Funds for the Central Universities (JBK1805001). Minghua Li’s work was supported in part by the National Natural Science Foundation of China (11301418), the Natural Science Foundation of Chongqing Municipal Science and Technology Commission (Grant Number: cstc2018jcyjAX0226), the Basic Science and Frontier Technology Research of Yongchuan (Grant Number: Ycstc, 2018nb1401), the Foundation for High-level Talents of Chongqing University of Art and Sciences (Grant Numbers: R2016SC13, P2017SC01), the Key Laboratory of Complex Data Analysis and Artificial Intelligence of Chongqing Municipal Science and Technology Commission. Xiaoqi Yang’s work was supported in part by the Research Grants Council of Hong Kong (PolyU 15218219).

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Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    K. W. Meng

  2. School of Mathematics and Big Data, Chongqing University of Arts and Sciences, Yongchuan, Chongqing, 402160, China

    M. H. Li

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    W. F. Yao & X. Q. Yang

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  1. K. W. Meng
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  2. M. H. Li
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  3. W. F. Yao
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  4. X. Q. Yang
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Correspondence to X. Q. Yang.

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This paper is dedicated to Professor Marco A. López on the occasion of his 70th birthday.

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Meng, K.W., Li, M.H., Yao, W.F. et al. Lipschitz-like property relative to a set and the generalized Mordukhovich criterion. Math. Program. 189, 455–489 (2021). https://doi.org/10.1007/s10107-020-01568-0

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