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

Author

Listed:
  • Duong Thi Kim Huyen

    (Vietnam Academy of Science and Technology)

  • Jen-Chih Yao

    (China Medical University)

  • Nguyen Dong Yen

    (Vietnam Academy of Science and Technology)

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).

Suggested Citation

  • Duong Thi Kim Huyen & Jen-Chih Yao & Nguyen Dong Yen, 2019. "Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 91-116, January.
    • Handle: RePEc:spr:joptap:v:180:y:2019:i:1:d:10.1007_s10957-018-1294-5
    DOI: 10.1007/s10957-018-1294-5
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    References listed on IDEAS

    as
    1. Jean-Pierre Aubin, 1984. "Lipschitz Behavior of Solutions to Convex Minimization Problems," Mathematics of Operations Research, INFORMS, vol. 9(1), pages 87-111, February.
    2. Nguyen Thanh Qui, 2014. "Generalized Differentiation of a Class of Normal Cone Operators," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 398-429, May.
    3. Nguyen Thanh Qui, 2012. "Nonlinear Perturbations of Polyhedral Normal Cone Mappings and Affine Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 98-122, April.
    4. Nguyen Thanh Qui, 2016. "Coderivatives of implicit multifunctions and stability of variational systems," Journal of Global Optimization, Springer, vol. 65(3), pages 615-635, July.
    5. B. S. Mordukhovich & T. T. A. Nghia & R. T. Rockafellar, 2015. "Full Stability in Finite-Dimensional Optimization," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 226-252, February.
    6. Shu Lu & Stephen M. Robinson, 2008. "Variational Inequalities over Perturbed Polyhedral Convex Sets," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 689-711, August.
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    Cited by:

    1. L. Q. Anh & T. Q. Duy & D. V. Hien, 2020. "Stability of efficient solutions to set optimization problems," Journal of Global Optimization, Springer, vol. 78(3), pages 563-580, November.

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