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Genericity and Hölder Stability in Semi-Algebraic Variational Inequalities

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Abstract

The aim of this paper is twofold. We first present generic properties of semi-algebraic variational inequalities: “typical” semi-algebraic variational inequalities have finitely many solutions, around each of which they admit a unique “active manifold” and such solutions are nondegenerate. Second, based on these results, we offer Hölder stability, upper semi-continuity, and lower semi-continuity properties of the solution map of parameterized variational inequalities.

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Acknowledgements

The authors would like to thank the referees for careful reading and constructive comments. A part of this work was done while the third author was visiting Department of Applied Mathematics, Pukyong National University, Busan, Korea, in September 2016. He would like to thank the department for the hospitality and support during his stay. The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2017R1E1A1A03069931). The third author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2016.05

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Authors and Affiliations

  1. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Republic of Korea

    Jae Hyoung Lee & Gue Myung Lee

  2. Department of Mathematics, University of Dalat, 1 Phu Dong Thien Vuong, Dalat, Vietnam

    Tiến-Sơn Phạm

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  1. Jae Hyoung Lee
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  2. Gue Myung Lee
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  3. Tiến-Sơn Phạm
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Correspondence to Gue Myung Lee.

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Communicated by Aris Daniilidis.

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Lee, J.H., Lee, G.M. & Phạm, TS. Genericity and Hölder Stability in Semi-Algebraic Variational Inequalities. J Optim Theory Appl 178, 56–77 (2018). https://doi.org/10.1007/s10957-018-1234-4

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  • DOI: https://doi.org/10.1007/s10957-018-1234-4

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