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The Stationary Point Set Map in General Parametric Optimization Problems

Abstract

The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953–986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples.

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Acknowledgements

The research of the first and the third authors is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), under grant number 101.01-2018.308. The second author was supported by the Grant MOST 105-2221-E-039-009-MY3 (Taiwan). The authors would like to thank the three anonymous referees for their very careful readings and valuable suggestions.

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Correspondence to D. T. K. Huyen.

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Huyen, D.T.K., Yao, JC. & Yen, N.D. The Stationary Point Set Map in General Parametric Optimization Problems. Set-Valued Var. Anal 30, 305–327 (2022). https://doi.org/10.1007/s11228-020-00557-x

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  • DOI: https://doi.org/10.1007/s11228-020-00557-x

Keywords

  • C 2-smooth optimization problem
  • Stationary point set
  • Total perturbation
  • Robinson stability
  • Lipschitz-like property
  • Second-order subdifferential
  • Coderivative

Mathematics Subject Classification (2010)

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