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Optimal Convergence Rates for Damped Inertial Gradient Dynamics with Flat Geometries

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A Correction to this article was published on 22 July 2023

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Abstract

In this paper, we are devoted to the study of a problem proposed in Aujol and Dossal (SIAM J Optim 29:3131–3153, 2019), which concerns the optimal convergence rates for damped inertial gradient dynamics with flat geometries. Fortunately, we find a solution to the problem. More specifically, we obtain an optimal decay rate for the system energy of the second-order differential equation with time-dependent damping in a Hilbert space \({\mathcal {H}}\)

$$\begin{aligned} \ddot{x}(t)+\gamma (t){\dot{x}}(t)+\nabla F(x(t))=0, \end{aligned}$$

where \(F : {\mathcal {H}} \rightarrow R\) is a differentiable convex function possessing at least one minimizer and \(\gamma (t)=\alpha /t\) with \(\alpha >0.\) It is known that this system, with the time-dependent damping coefficient \(\gamma (t)\), models a nonlinear oscillator with viscous damping, and is associated with the Nesterov acceleration scheme, FISTA, or the accelerated gradient method in the area of numerical optimization.

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Acknowledgements

The authors would like to thank the referee very much for his/her professional and valuable comments and suggestions. The work was supported partly by the NSF of China (Grant Numbers 11947004, 11831011) and the Shanghai Key Laboratory for Contemporary Applied Mathematics (Grant Number 08DZ2271900).

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Jun-Ren Luo

  2. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Ti-Jun Xiao

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  1. Jun-Ren Luo
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Luo, JR., Xiao, TJ. Optimal Convergence Rates for Damped Inertial Gradient Dynamics with Flat Geometries. Appl Math Optim 87, 53 (2023). https://doi.org/10.1007/s00245-023-09966-4

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