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.
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Acknowledgements
The authors are very grateful to the anonymous reviewers for their suggestions and comments that improved the presentation of this paper. This research was supported by the National Natural Science Foundations of China (Grant 11971422), CAS “Light of West China” Program, the Natural Science Foundation of Yunnan Province (No. 2018FB004) and the Scientific Research Foundation of Yunnan University under Grant No. 2018YDJQ010, and by Joint Key Project of Yunnan Provincial Science and Technology Department and Yunnan University (No. 2018FY001(-014)) and IRTSTYN. The research of Professor Jen-Chih Yao was supported by the Grant MOST 108-2115-M-039-005-MY3.
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Communicated by René Henrion.
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Wei, Z., Tammer, C. & Yao, JC. Characterizations for Strong Abadie Constraint Qualification and Applications to Calmness. J Optim Theory Appl 189, 1–18 (2021). https://doi.org/10.1007/s10957-020-01808-5
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DOI: https://doi.org/10.1007/s10957-020-01808-5
Keywords
- Strong Abadie constraint qualification
- Abadie constraint qualification
- Convex multifunction
- Calmness
- Error bounds