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On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms

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  • F. Lara

    (Universidad de Tarapacá)

Abstract

We prove that every strongly quasiconvex function is 2-supercoercive (in particular, coercive). Furthermore, we investigate the usual properties of proximal operators for strongly quasiconvex functions. In particular, we prove that the set of fixed points of the proximal operator coincides with the unique minimizer of a lower semicontinuous strongly quasiconvex function. As a consequence, we implement the proximal point algorithm for finding the unique solution of the minimization problem of a strongly quasiconvex function by using a positive sequence of parameters bounded away from 0 and, in particular, we revisit the general quasiconvex case. Moreover, a new characterization for convex functions is derived from this analysis. Finally, an application for a strongly quasiconvex function which is neither convex nor differentiable nor locally Lipschitz continuous is provided.

Suggested Citation

  • F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.
    • Handle: RePEc:spr:joptap:v:192:y:2022:i:3:d:10.1007_s10957-021-01996-8
    DOI: 10.1007/s10957-021-01996-8
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    References listed on IDEAS

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    8. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
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    Cited by:

    1. Sorin-Mihai Grad & Felipe Lara & Raúl Tintaya Marcavillaca, 2024. "Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2233-2262, December.
    2. A. Kabgani & F. Lara, 2022. "Strong subdifferentials: theory and applications in nonconvex optimization," Journal of Global Optimization, Springer, vol. 84(2), pages 349-368, October.
    3. S.-M. Grad & F. Lara & R. T. Marcavillaca, 2023. "Relaxed-inertial proximal point type algorithms for quasiconvex minimization," Journal of Global Optimization, Springer, vol. 85(3), pages 615-635, March.
    4. Chinedu Izuchukwu & Grace N. Ogwo & Yekini Shehu, 2024. "Proximal Point Algorithms with Inertial Extrapolation for Quasi-convex Pseudo-monotone Equilibrium Problems," Networks and Spatial Economics, Springer, vol. 24(3), pages 681-706, September.
    5. Erik Alex Papa Quiroz, 2024. "Proximal Point Method for Quasiconvex Functions in Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1268-1285, September.
    6. Alfredo Iusem & Felipe Lara & Raúl T. Marcavillaca & Le Hai Yen, 2024. "A Two-Step Proximal Point Algorithm for Nonconvex Equilibrium Problems with Applications to Fractional Programming," Journal of Global Optimization, Springer, vol. 90(3), pages 755-779, November.
    7. Tran Thang & Xuan Thanh Le, 2024. "Self-Adaptive Extragradient Algorithms for Quasi-Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2988-3013, December.

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