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Fast Convergence of Inertial Gradient Dynamics with Multiscale Aspects

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  • Haixin Ren

    (Harbin Engineering University)

  • Bin Ge

    (Harbin Engineering University)

  • Xiangwu Zhuge

    (Harbin Engineering University)

Abstract

In this paper, the asymptotic properties as $$ t\rightarrow +\infty $$ t → + ∞ of the following second-order differential equation in a Hilbert space $$ {\mathcal {H}} $$ H are studied, $$\begin{aligned} \ddot{x}(t)+\gamma (t){\dot{x}}(t)+\beta (t)\Big (\nabla \Phi (x(t))+\epsilon (t)\nabla U(x(t))\Big )=0, \end{aligned}$$ x ¨ ( t ) + γ ( t ) x ˙ ( t ) + β ( t ) ( ∇ Φ ( x ( t ) ) + ϵ ( t ) ∇ U ( x ( t ) ) ) = 0 , where $$ \Phi ,U:{\mathcal {H}}\rightarrow {\mathbb {R}} $$ Φ , U : H → R are convex differentiable, $$ \gamma (t) $$ γ ( t ) is a positive damping coefficient, $$ \beta (t) $$ β ( t ) is a time scale coefficient and $$ \epsilon (t) $$ ϵ ( t ) is a positive nonincreasing function, $$\gamma (t)$$ γ ( t ) , $$ \beta (t) $$ β ( t ) and $$ \epsilon (t) $$ ϵ ( t ) are all continuously differentiable. This system has applications in the fields of mechanics and optimization. Based on the proper tuning of $$ \gamma (t) $$ γ ( t ) and $$ \beta (t) $$ β ( t ) , we obtain the convergence rates for the values, and the conclusion is that, under the different conditions, the trajectories either converge to one minimizer of $$ \Phi $$ Φ weakly, or converge to one common minimizer of $$ \Phi $$ Φ and U weakly. When $$ \epsilon (t) $$ ϵ ( t ) tends to 0 as t goes to infinity, under the condition that $$ \Phi $$ Φ or U is convex, the trajectories converge to the unique minimizer of $$ \Phi $$ Φ , or the unique minimizer of U, respectively. Finally, some particular cases are examined, and some numerical experiments are conducted to illustrate our main results.

Suggested Citation

  • Haixin Ren & Bin Ge & Xiangwu Zhuge, 2023. "Fast Convergence of Inertial Gradient Dynamics with Multiscale Aspects," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 461-489, February.
  • Handle: RePEc:spr:joptap:v:196:y:2023:i:2:d:10.1007_s10957-022-02124-w
    DOI: 10.1007/s10957-022-02124-w
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. 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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