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Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems

Author

Listed:
  • R. S. Burachik

    (University of South Australia)

  • X. Q. Yang

    (The Hong Kong Polytechnic University)

  • Y. Y. Zhou

    (Soochow University)

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.

Suggested Citation

  • R. S. Burachik & X. Q. Yang & Y. Y. Zhou, 2017. "Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 471-503, May.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-017-1091-6
    DOI: 10.1007/s10957-017-1091-6
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    References listed on IDEAS

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    1. Alexander Shapiro & Jie Sun, 2004. "Some Properties of the Augmented Lagrangian in Cone Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 479-491, August.
    2. X. X. Huang & X. Q. Yang, 2003. "A Unified Augmented Lagrangian Approach to Duality and Exact Penalization," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 533-552, August.
    3. S. Wang & X. Q. Yang & K. L. Teo, 2006. "Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation," Journal of Optimization Theory and Applications, Springer, vol. 129(2), pages 227-254, May.
    4. Yu Zhou & Jin Zhou & Xiao Yang, 2014. "Existence of augmented Lagrange multipliers for cone constrained optimization problems," Journal of Global Optimization, Springer, vol. 58(2), pages 243-260, February.
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    Cited by:

    1. M. V. Dolgopolik, 2018. "Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property," Journal of Global Optimization, Springer, vol. 71(2), pages 237-296, June.

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