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Characterizations for Strong Abadie Constraint Qualification and Applications to Calmness

Author

Listed:
  • Zhou Wei

    (Yunnan University)

  • Christiane Tammer

    (Martin-Luther-University Halle-Wittenberg)

  • Jen-Chih Yao

    (China Medical University
    National Sun Yat-Sen University)

Abstract

In this paper, we mainly study the Abadie constraint qualification (ACQ) and the strong ACQ of a convex multifunction. To characterize the general difference between strong ACQ and ACQ, we prove that the strong ACQ is essentially equivalent to the ACQ plus the finite distance of two image sets of the tangent derivative multifunction on the sphere and the origin, respectively. This characterization for the strong ACQ is used to provide the exact calmness modulus of a convex multifunction. Finally, we apply these results to local and global error bounds for a convex inequality defined by a proper convex function. The characterization of the strong ACQ enables us to give primal equivalent criteria for local and global error bounds in terms of contingent cones and directional derivatives.

Suggested Citation

  • Zhou Wei & Christiane Tammer & Jen-Chih Yao, 2021. "Characterizations for Strong Abadie Constraint Qualification and Applications to Calmness," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 1-18, April.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:1:d:10.1007_s10957-020-01808-5
    DOI: 10.1007/s10957-020-01808-5
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    References listed on IDEAS

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    1. M. J. Cánovas & R. Henrion & M. A. López & J. Parra, 2016. "Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 925-952, June.
    2. Zhou Wei & Jen-Chih Yao, 2017. "Abadie Constraint Qualifications for Convex Constraint Systems and Applications to Calmness Property," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 388-407, August.
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