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On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization

Author

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  • Gulcin Dinc Yalcin

    (Eskisehir Technical University)

  • Refail Kasimbeyli

    (Eskisehir Technical University)

Abstract

In this paper, zero duality gap conditions in nonconvex optimization are investigated. It is considered that dual problems can be constructed with respect to the weak conjugate functions, and/or directly by using an augmented Lagrangian formulation. Both of these approaches and the related strong duality theorems are studied and compared in this paper. By using the weak conjugate functions approach, special cases related to the optimization problems with equality and inequality constraints are studied and the zero duality gap conditions in terms of objective and constraint functions, are established. Illustrative examples are provided.

Suggested Citation

  • Gulcin Dinc Yalcin & Refail Kasimbeyli, 2020. "On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 199-228, August.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:1:d:10.1007_s00186-020-00708-8
    DOI: 10.1007/s00186-020-00708-8
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    References listed on IDEAS

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    1. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
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    3. Emil Ernst & Michel Volle, 2013. "Zero Duality Gap for Convex Programs: A Generalization of the Clark–Duffin Theorem," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 668-686, September.
    4. 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.
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    6. B. T. Polyak, 1998. "Convexity of Quadratic Transformations and Its Use in Control and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 553-583, December.
    7. 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.
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