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Fast Convex Optimization via Differential Equation with Hessian-Driven Damping and Tikhonov Regularization

Author

Listed:
  • Gangfan Zhong

    (South China Normal University
    Beijing University of Technology)

  • Xiaozhe Hu

    (Tufts University)

  • Ming Tang

    (South China Normal University
    Guangzhou University)

  • Liuqiang Zhong

    (South China Normal University)

Abstract

In this paper, we consider a class of second-order ordinary differential equations with Hessian-driven damping and Tikhonov regularization, which arises from the minimization of a smooth convex function in Hilbert spaces. Inspired by Attouch et al. (J Differ Equ 261:5734–5783, 2016), we establish that the function value along the solution trajectory converges to the optimal value, and prove that the convergence rate can be as fast as $$o(1/t^2)$$ o ( 1 / t 2 ) . By constructing proper energy function, we prove that the trajectory strongly converges to a minimizer of the objective function of minimum norm. Moreover, we propose a gradient-based optimization algorithm based on numerical discretization, and demonstrate its effectiveness in numerical experiments.

Suggested Citation

  • Gangfan Zhong & Xiaozhe Hu & Ming Tang & Liuqiang Zhong, 2024. "Fast Convex Optimization via Differential Equation with Hessian-Driven Damping and Tikhonov Regularization," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 42-82, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02462-x
    DOI: 10.1007/s10957-024-02462-x
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