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Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 91–116 | Cite as

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability

  • Duong Thi Kim Huyen
  • Jen-Chih Yao
  • Nguyen Dong YenEmail author
Article

Abstract

By applying some theorems of Levy and Mordukhovich (Math Program 99:311–327, 2004) and other related results, we estimate the Fréchet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From the obtained formulas, we derive necessary and sufficient conditions for the local Lipschitz-like property of the stationary point set map. This leads us to new insights into the preceding deep investigations of Levy and Mordukhovich in the above-cited paper and of Qui (J Optim Theory Appl 161:398–429, 2014, J Glob Optim 65:615–635, 2016).

Keywords

Smooth parametric optimization problem Smooth functional constraint Stationary point set map Lipschitz-like property Coderivative 

Mathematics Subject Classification

49K40 49J53 90C31 90C20 

Notes

Acknowledgements

This work was supported by National Foundation for Science & Technology Development (Vietnam) and the Grant MOST 105-2115-M-039-002-MY3 (Taiwan). The authors are grateful to the anonymous referees for their careful readings, encouragement, and valuable suggestions. Examples 3.4 and 4.2 in this paper present our solutions to two open questions raised by one of the referees. In addition, Remarks 3.2 and 3.3 are based on some comments of that referee.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Duong Thi Kim Huyen
    • 1
  • Jen-Chih Yao
    • 2
  • Nguyen Dong Yen
    • 3
    Email author
  1. 1.Graduate Training Center, Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Center for General EducationChina Medical UniversityTaichungTaiwan
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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