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New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization

Author

Listed:
  • C. Gutiérrez

    (Universidad de Valladolid)

  • B. Jiménez

    (Universidad Nacional de Educación a Distancia)

  • V. Novo

    (Universidad Nacional de Educación a Distancia)

Abstract

We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are studied.

Suggested Citation

  • C. Gutiérrez & B. Jiménez & V. Novo, 2009. "New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 85-106, July.
    • Handle: RePEc:spr:joptap:v:142:y:2009:i:1:d:10.1007_s10957-009-9525-4
    DOI: 10.1007/s10957-009-9525-4
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    References listed on IDEAS

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    1. B. Aghezzaf & M. Hachimi, 1999. "Second-Order Optimality Conditions in Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 37-50, July.
    2. P. Q. Khanh & N. D. Tuan, 2006. "First and Second Order Optimality Conditions Using Approximations for Nonsmooth Vector Optimization in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 289-308, August.
    3. Giancarlo Bigi, 2006. "On sufficient second order optimality conditions in multiobjective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 77-85, February.
    4. Bienvenido Jiménez & Vicente Novo, 2003. "Second order necessary conditions in set constrained differentiable vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 299-317, November.
    5. M. Hachimi & B. Aghezzaf, 2007. "New Results on Second-Order Optimality Conditions in Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 135(1), pages 117-133, October.
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    Cited by:

    1. Nguyen Minh Tung & Nguyen Xuan Duy Bao, 2023. "New Set-Valued Directional Derivatives: Calculus and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 411-437, May.
    2. Jinchuan Zhou & Jingyong Tang & Jein-Shan Chen, 2017. "Parabolic Second-Order Directional Differentiability in the Hadamard Sense of the Vector-Valued Functions Associated with Circular Cones," Journal of Optimization Theory and Applications, Springer, vol. 172(3), pages 802-823, March.
    3. S. J. Li & S. K. Zhu & X. B. Li, 2012. "Second-Order Optimality Conditions for Strict Efficiency of Constrained Set-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 534-557, November.
    4. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.

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