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Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems

Author

Listed:
  • Lien T. Nguyen

    (RMIT University)

  • Andrew Eberhard

    (RMIT University)

  • Xinghuo Yu

    (RMIT University)

  • Alexander Y. Kruger

    (Ton Duc Thang University)

  • Chaojie Li

    (University of New South Wales)

Abstract

In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka–Łojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence.

Suggested Citation

  • Lien T. Nguyen & Andrew Eberhard & Xinghuo Yu & Alexander Y. Kruger & Chaojie Li, 2024. "Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 844-879, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02536-w
    DOI: 10.1007/s10957-024-02536-w
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    References listed on IDEAS

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    1. Sedi Bartz & Minh N. Dao & Hung M. Phan, 2022. "Conical averagedness and convergence analysis of fixed point algorithms," Journal of Global Optimization, Springer, vol. 82(2), pages 351-373, February.
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, December.
    3. B. Abbas & H. Attouch & Benar F. Svaiter, 2014. "Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 331-360, May.
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