Hölder Error Bounds and Hölder Calmness with Applications to Convex Semi-infinite Optimization

Abstract

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.

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Acknowledgements

The authors would like to thank Prof. Xi Yin Zheng for his valuable comments which helped us improve the presentation of the paper and Dr Li Minghua who has read the preliminary version of the manuscript and found a mistake in one of the proofs. We are grateful to PhD students Bui Thi Hoa and Nguyen Duy Cuong from Federation University Australia for carefully reading the manuscript, eliminating numerous glitches and contributing to Example 3.19.

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Correspondence to Alexander Y. Kruger.

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The research of the first and second authors was supported by the Australian Research Council, project DP160100854. The first author benefited from the support of the FMJH Program PGMO and from the support of EDF. The research of the second author was also supported by MINECO of Spain and FEDER of EU, grant MTM2014-59179-C2-1-P. The research of the third author was supported by the Research Grants Council of Hong Kong (PolyU 152342/16E). The research of the fourth author was supported by the National Natural Science Foundation of China, grants 11771384 and 11461080.

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Kruger, A.Y., López, M.A., Yang, X. et al. Hölder Error Bounds and Hölder Calmness with Applications to Convex Semi-infinite Optimization. Set-Valued Var. Anal 27, 995–1023 (2019). https://doi.org/10.1007/s11228-019-0504-0

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Keywords

  • Hölder error bounds
  • Hölder calmness
  • Convex programming
  • Semi-infinite programming

Mathematics Subject Classification (2010)

  • 49J53
  • 90C25
  • 90C31
  • 90C34