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Unified Nonlinear Lagrangian Approach to Duality and Optimal Paths

Author

Listed:
  • C. Y. Wang

    (Qufu Normal University)

  • X. Q. Yang

    (Hong Kong Polytechnic University)

  • X. M. Yang

    (Chongqing Normal University)

Abstract

In the context of an inequality constrained optimization problem, we present a unified nonlinear Lagrangian dual scheme and establish necessary and sufficient conditions for the zero duality gap property. From these results, we derive necessary and sufficient conditions for four classes of zero duality gap properties and establish the equivalence among them. Finally, we obtain the convergence of an optimal path for the unified scheme and present a sufficient condition for the finite termination of the optimal path.

Suggested Citation

  • C. Y. Wang & X. Q. Yang & X. M. Yang, 2007. "Unified Nonlinear Lagrangian Approach to Duality and Optimal Paths," Journal of Optimization Theory and Applications, Springer, vol. 135(1), pages 85-100, October.
    • Handle: RePEc:spr:joptap:v:135:y:2007:i:1:d:10.1007_s10957-007-9225-x
    DOI: 10.1007/s10957-007-9225-x
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    References listed on IDEAS

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    1. A. M. Rubinov & X. X. Huang & X. Q. Yang, 2002. "The Zero Duality Gap Property and Lower Semicontinuity of the Perturbation Function," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 775-791, November.
    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. C. J. Goh & X. Q. Yang, 2001. "Nonlinear Lagrangian Theory for Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 99-121, April.
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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.
    2. M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 728-744, March.
    3. R. S. Burachik & A. N. Iusem & J. G. Melo, 2010. "Duality and Exact Penalization for General Augmented Lagrangians," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 125-140, October.
    4. 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.
    5. 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.
    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.

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