Asymptotic for a second order evolution equation with convex potential and vanishing damping term

By Ramzi May. 2015

106 citations recorded by the source corpus.

In this short note, we recover by a different method the new result due to Attouch, Peyrouqet and Redont concerning the weak convergence as $t\rightarrow+\infty$ of solutions $x(t)$ to the second order differential equation \[ x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nabla\Phi(x(t))=0, \] where $K>3$ and $\Phi$ is a smooth convex function defined on an Hilbert Space $\mathcal{H}.$ Moreover, we improve slightly their result on the rate of convergence of $\Phi(x(t))-\min\Phi.$

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/a3eb7cedc5fd6db953d240173e046d0be95d1a20

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In Applied Mathematics

In Computational Theory and Mathematics

In Modeling and Simulation

Citations in the corpus

Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping (2021, background): “For such values of α, it was also proved in [17] and [29] that the asymptotic convergence rate of the values is actually o(1/t2).”
Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3 (2017, methodology): “OC] 5 May 2015.”
Fast optimization via inertial dynamics with closed-loop damping (2020, background): “It is shown in [33] and [78] that the asymptotic convergence rate of the values is actually o(1/t).”