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Normal Cone and Subdifferential with Respect to a Set at Infinity and Their Applications

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  • Nguyen Le Hoang Anh

    (University of Science
    Vietnam National University)

  • Nguyen Canh Hung

    (University of Science
    Vietnam National University)

Abstract

In this paper, we first establish some properties of normal cones with respect to a set and derive calculus rules for subdifferentials with respect to a set at a reference point. Next, we introduce the concepts of normal cone with respect to a set to an epigraph at infinity, and subdifferential with respect to a set at infinity. Then, several properties of these notions at infinity are explored. Finally, we examine these tools at infinity to obtain necessary optimality conditions at infinity, the compactness of the optimal solution set, and the coercivity of the objective function for the underlying optimization problem under the unboundedness of its associated feasible set.

Suggested Citation

  • Nguyen Le Hoang Anh & Nguyen Canh Hung, 2025. "Normal Cone and Subdifferential with Respect to a Set at Infinity and Their Applications," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-27, September.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:3:d:10.1007_s10957-025-02761-x
    DOI: 10.1007/s10957-025-02761-x
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    References listed on IDEAS

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    1. F. Flores-Bazán & N. Hadjisavvas & F. Lara & I. Montenegro, 2016. "First- and Second-Order Asymptotic Analysis with Applications in Quasiconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 372-393, August.
    2. E. Allevi & J. E. Martínez-Legaz & R. Riccardi, 2020. "Optimality conditions for convex problems on intersections of non necessarily convex sets," Journal of Global Optimization, Springer, vol. 77(1), pages 143-155, May.
    3. César Gutiérrez & Rubén López & Vicente Novo, 2014. "Existence and Boundedness of Solutions in Infinite-Dimensional Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 515-547, August.
    4. Felipe Lara & Rubén López, 2017. "Formulas for Asymptotic Functions via Conjugates, Directional Derivatives and Subdifferentials," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 793-811, June.
    5. Fabián Flores-Bazán & Fernando Flores-Bazán & Cristián Vera, 2015. "Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives," Journal of Global Optimization, Springer, vol. 63(1), pages 99-123, September.
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