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Strong Abadie CQ, ACQ, calmness and linear regularity

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Abstract

The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relationships among ACQ, strong ACQ, basic constraint qualification (BCQ) and strong BCQ are discussed. Using the strong ACQ, we study calmness of a closed and convex multifunction between two Banach spaces and, different from many existing dual conditions for calmness, establish several primal characterizations of calmness. As applications, some primal characterizations for error bounds and linear regularity are established; in particular, some existing results are improved.

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Acknowledgments

The authors wish to thank the referee for careful reading of the paper and helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, People’s Republic of China

    Zhou Wei & Xi Yin Zheng

  2. Center for General Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan

    Jen-Chih Yao

  3. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

    Jen-Chih Yao

Authors
  1. Zhou Wei
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  2. Jen-Chih Yao
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  3. Xi Yin Zheng
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Correspondence to Xi Yin Zheng.

Additional information

This research was supported by the National Natural Science Foundation of P. R. China (Grant No. 11061038, No. 11061039), IRTSTYN and the Grant NSC 99-2115-M-110-004-MY3.

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Wei, Z., Yao, JC. & Zheng, X.Y. Strong Abadie CQ, ACQ, calmness and linear regularity. Math. Program. 145, 97–131 (2014). https://doi.org/10.1007/s10107-013-0641-4

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  • DOI: https://doi.org/10.1007/s10107-013-0641-4

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