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Second-Order Set-Valued Directional Derivatives of the Marginal Map in Parametric Vector Optimization Problems

Author

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  • Nguyen Xuan Duy Bao

    (University of Technology
    Vietnam National University)

  • Phan Quoc Khanh

    (Ton Duc Thang University)

  • Nguyen Minh Tung

    (Ho Chi Minh University of Banking)

Abstract

We study second-order differential sensitivity in parametrized vector optimization problems with inclusion constraints. First, we consider a set-valued unconstrained problem and establish a sufficient condition for the second-order directional Dini derivative of the marginal map to be equal to the minimum of that of the objective map. We then extend our research to vector optimization problems with general inclusion constraints and demonstrate that the first- and second-order directional Dini derivatives of the objective image map are equal to the union of those of the objective map. Using advanced proof techniques, we derive a formula for the second-order directional Dini derivative of the marginal map and prove the second-order semi-derivability of the feasible objective and marginal/efficient-value maps. Examples are provided to illustrate the novelty and depth of our results.

Suggested Citation

  • Nguyen Xuan Duy Bao & Phan Quoc Khanh & Nguyen Minh Tung, 2025. "Second-Order Set-Valued Directional Derivatives of the Marginal Map in Parametric Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 204(3), pages 1-20, March.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-025-02606-7
    DOI: 10.1007/s10957-025-02606-7
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    References listed on IDEAS

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    1. Tuan, Nguyen Dinh, 2015. "First and second-order optimality conditions for nonsmooth vector optimization using set-valued directional derivatives," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 300-317.
    2. P. Q. Khanh & N. D. Tuan, 2007. "Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 341-357, June.
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