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

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).

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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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Correspondence to Nguyen Dong Yen.

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Huyen, D.T.K., Yao, J. & Yen, N.D. Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability. J Optim Theory Appl 180, 91–116 (2019). https://doi.org/10.1007/s10957-018-1294-5

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Keywords

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

Mathematics Subject Classification

  • 49K40
  • 49J53
  • 90C31
  • 90C20