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A Continuity Result for the Adjusted Normal Cone Operator

Author

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  • Marco Castellani

    (Computer Science and Mathematics)

  • Massimiliano Giuli

    (Computer Science and Mathematics)

Abstract

The concept of adjusted sublevel set for a quasiconvex function was introduced by Aussel and Hadjisavvas who proved the local existence of a norm-to-weak $$^*$$ ∗ upper semicontinuous base-valued submap of the normal operator associated with the adjusted sublevel set. When the space is finite-dimensional, a globally defined upper semicontinuous base-valued submap is obtained by taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique does not work in the infinite-dimensional case. We propose a partition of unity technique to overcome this problem in Banach spaces. An application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem.

Suggested Citation

  • Marco Castellani & Massimiliano Giuli, 2024. "A Continuity Result for the Adjusted Normal Cone Operator," Journal of Optimization Theory and Applications, Springer, vol. 200(2), pages 858-873, February.
    • Handle: RePEc:spr:joptap:v:200:y:2024:i:2:d:10.1007_s10957-023-02326-w
    DOI: 10.1007/s10957-023-02326-w
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    References listed on IDEAS

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    1. Suliman Al-Homidan & Nicolas Hadjisavvas & Loai Shaalan, 2018. "Transformation of Quasiconvex Functions to Eliminate Local Minima," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 93-105, April.
    2. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    3. D. Aussel & J. Cotrina, 2013. "Quasimonotone Quasivariational Inequalities: Existence Results and Applications," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 637-652, September.
    4. Mircea Balaj & Marco Castellani & Massimiliano Giuli, 2023. "New criteria for existence of solutions for equilibrium problems," Computational Management Science, Springer, vol. 20(1), pages 1-16, December.
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