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Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems

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Abstract

Using an augmented Lagrangian approach, we study the existence of augmented Lagrange multipliers of a semi-infinite programming problem and discuss their characterizations in terms of saddle points. In the case of a sharp Lagrangian, we obtain a first-order necessary condition for the existence of an augmented Lagrange multiplier for the semi-infinite programming problem and some first-order sufficient conditions by assuming inf-compactness of the data functions and the extended Mangasarian–Fromovitz constraint qualification. Using a valley at 0 augmenting function and assuming suitable second-order sufficient conditions, we obtain the existence of an augmented Lagrange multiplier for the semi-infinite programming problem.

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Acknowledgements

The authors are grateful to the referees and the associate Editor for their careful reading and comments which have improved the final presentation of the paper. The work described in this paper was partially supported by Grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (PolyU 5334/08E and PolyU 5292/13E) and by Natural Science Foundation of China (11471235, 11171247 and 11371273).

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Authors and Affiliations

  1. School of Information Technology and Mathematical Sciences, University of South Australia, Adelaide, Australia

    R. S. Burachik

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    X. Q. Yang

  3. Department of Mathematics, Soochow University, Suzhou, 215006, China

    Y. Y. Zhou

Authors
  1. R. S. Burachik
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  2. X. Q. Yang
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  3. Y. Y. Zhou
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Corresponding author

Correspondence to X. Q. Yang.

Additional information

Communicated by Marco Antonio López-Cerdá.

This paper is dedicated to the memory of the late Professor Vladimir Demyanov.

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Burachik, R.S., Yang, X.Q. & Zhou, Y.Y. Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems. J Optim Theory Appl 173, 471–503 (2017). https://doi.org/10.1007/s10957-017-1091-6

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  • DOI: https://doi.org/10.1007/s10957-017-1091-6

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