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

Gangfan Zhong (), Xiaozhe Hu (), Ming Tang () and Liuqiang Zhong ()
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Gangfan Zhong: South China Normal University
Xiaozhe Hu: Tufts University
Ming Tang: South China Normal University
Liuqiang Zhong: South China Normal University

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 1, No 3, 42-82

Abstract: 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.

Keywords: Convex optimization; Second order ordinary equation; Hessian-driven damping; Tikhonov regularization; 37N40; 46N10; 65B99; 65K05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02462-x

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