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Characterizations, Dynamical Systems and Gradient Methods for Strongly Quasiconvex Functions

Author

Listed:
  • Felipe Lara

    (Universidad de Tarapacá)

  • Raúl T. Marcavillaca

    (Universidad de Chile)

  • Phan Tu Vuong

    (University of Southampton
    HCMC University of Technology and Education)

Abstract

We study differentiable strongly quasiconvex functions for providing new properties for algorithmic and monotonicity purposes. Furthermore, we provide insights into the decreasing behaviour of strongly quasiconvex functions, applying this for establishing exponential convergence for first- and second-order gradient systems without relying on the usual Lipschitz continuity assumption on the gradient of the function. The explicit discretization of the first-order dynamical system leads to the gradient descent method while discretization of the second-order dynamical system with viscous damping recovers the heavy ball method. We establish the linear convergence of both methods under suitable conditions on the parameters as well as numerical experiments for supporting our theoretical findings.

Suggested Citation

  • Felipe Lara & Raúl T. Marcavillaca & Phan Tu Vuong, 2025. "Characterizations, Dynamical Systems and Gradient Methods for Strongly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-25, September.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:3:d:10.1007_s10957-025-02728-y
    DOI: 10.1007/s10957-025-02728-y
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    References listed on IDEAS

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    1. Mohsen Rahimi Piranfar & Hadi Khatibzadeh, 2021. "Long-Time Behavior of a Gradient System Governed by a Quasiconvex Function," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 169-191, January.
    2. 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.
    3. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, April.
    4. Ion Necoara & Yurii Nesterov & François Glineur, 2019. "Linear convergence of first order methods for non-strongly convex optimization," LIDAM Reprints CORE 3000, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. 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.
    6. J. Bolte, 2003. "Continuous Gradient Projection Method in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 235-259, November.
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