Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 117–139 | Cite as

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 2: Robinson Stability

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


In Part 1 of this paper, we have estimated 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 these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.


Smooth parametric optimization problem Smooth functional constraint Stationary point set map Robinson stability Coderivative 

Mathematics Subject Classification

49K40 49J53 90C31 90C20 



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. Section 5 is based on the comments made by one of the referees.


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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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