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Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

Author

Listed:
  • Hezhi Luo

    (Zhejiang University of Technology)

  • Huixian Wu

    (Hangzhou Dianzi University)

  • Jianzhen Liu

    (Hangzhou Dianzi University)

Abstract

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.

Suggested Citation

  • Hezhi Luo & Huixian Wu & Jianzhen Liu, 2013. "Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 360-385, November.
    • Handle: RePEc:spr:joptap:v:159:y:2013:i:2:d:10.1007_s10957-013-0358-9
    DOI: 10.1007/s10957-013-0358-9
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    References listed on IDEAS

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    1. H. Wu & H. Luo, 2012. "Saddle points of general augmented Lagrangians for constrained nonconvex optimization," Journal of Global Optimization, Springer, vol. 53(4), pages 683-697, August.
    2. Gianni Di Pillo & Stefano Lucidi & Laura Palagi, 2005. "Convergence to Second-Order Stationary Points of a Primal-Dual Algorithm Model for Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 897-915, November.
    3. H. Z. Luo & G. Mastroeni & H. X. Wu, 2010. "Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 275-290, February.
    4. X. L. Sun & H. Z. Luo & D. Li, 2007. "Convexification of Nonsmooth Monotone Functions1," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 339-351, February.
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    Cited by:

    1. Hezhi Luo & Huixian Wu & Jianzhen Liu, 2015. "On Saddle Points in Semidefinite Optimization via Separation Scheme," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 113-150, April.
    2. Shengjie Li & Yangdong Xu & Manxue You & Shengkun Zhu, 2018. "Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 609-636, June.
    3. 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.
    4. M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 745-762, March.
    5. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 738-762, June.
    6. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part II: Special Duality Schemes," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 763-782, June.
    7. Manxue You & Shengjie Li, 2017. "Separation Functions and Optimality Conditions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 527-544, November.

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