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Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization

Author

Listed:
  • Nguyen Xuan Duy Bao

    (University of Science
    Vietnam National University)

  • Phan Quoc Khanh

    (Ton Duc Thang University)

  • Nguyen Minh Tung

    (Banking University of Ho Chi Minh City)

Abstract

We consider higher-order conditions and sensitivity analysis for solutions to equilibrium problems. The conditions for solutions are in terms of quasi-contingent derivatives and involve higher-order complementarity slackness for both the objective and the constraints and under Hölder metric subregularity assumptions. For sensitivity analysis, a formula of this type of derivative of the solution map to a parametric equilibrium problem is established in terms of the same types of derivatives of the data of the problem. Here, the concepts of a quasi-contingent derivative and critical directions are new. We consider open-cone solutions and proper solutions. We also study an important and typical special case: weak solutions of a vector minimization problem with mixed constraints. The results are significantly new and improve recent corresponding results in many aspects.

Suggested Citation

  • Nguyen Xuan Duy Bao & Phan Quoc Khanh & Nguyen Minh Tung, 2022. "Quasi-contingent derivatives and studies of higher-orders in nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(1), pages 205-228, September.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:1:d:10.1007_s10898-022-01129-z
    DOI: 10.1007/s10898-022-01129-z
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    References listed on IDEAS

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    1. P. Q. Khanh & N. M. Tung, 2015. "Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 68-90, October.
    2. S. J. Li & K. L. Teo & X. Q. Yang, 2008. "Higher-Order Optimality Conditions for Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 533-553, June.
    3. N. L. H. Anh & P. Q. Khanh, 2013. "Variational Sets of Perturbation Maps and Applications to Sensitivity Analysis for Constrained Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 363-384, August.
    4. Boris Mordukhovich & Wei Ouyang, 2015. "Higher-order metric subregularity and its applications," Journal of Global Optimization, Springer, vol. 63(4), pages 777-795, December.
    5. Phan Khanh & Le Tung, 2013. "First and second-order optimality conditions using approximations for vector equilibrium problems with constraints," Journal of Global Optimization, Springer, vol. 55(4), pages 901-920, April.
    6. Nguyen Hoang Anh & Phan Khanh, 2014. "Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives," Journal of Global Optimization, Springer, vol. 58(4), pages 693-709, April.
    7. E. K. Makarov & N. N. Rachkovski, 1999. "Unified Representation of Proper Efficiency by Means of Dilating Cones," Journal of Optimization Theory and Applications, Springer, vol. 101(1), pages 141-165, April.
    8. P. Q. Khanh & N. D. Tuan, 2008. "Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 243-261, November.
    9. H. T. H. Diem & P. Q. Khanh & L. T. Tung, 2014. "On Higher-Order Sensitivity Analysis in Nonsmooth Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 463-488, August.
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