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Pré-Publication, Document De Travail Année : 2023

A doubly nonlinear evolution system with threshold effects associated with dry friction

Samir Adly
Hedy Attouch
Manh Hung Le
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Résumé

In this paper, we investigate the asymptotic behavior of inertial dynamics with dry friction within the context of a Hilbert framework for convex differentiable optimization. Our study focuses on a doubly nonlinear first-order evolution inclusion that encompasses two potentials. In our analysis, we specifically focus on two main components: the differentiable function f that needs to be minimized, which influences the system's state through its gradient, and the nonsmooth dry friction potential denoted as ϕ = r •. It's important to note that the dry friction term acts on a linear combination of the velocity vector and the gradient of f. Consequently, any stationary point in our system corresponds to a critical point of f , unlike the case where only the velocity vector is involved in the dry friction term, resulting in an approximate critical point of f. To emphasize the crucial role of ∇f (x), we also explore the dual formulation of this dynamic, which possesses a Riemannian gradient structure. To address these dynamics, we employ the recently developed generic acceleration approach by Attouch, Bot, and Nguyen. This approach involves the time scaling of a continuous first-order differential equation, followed by the application of the method of averaging. By applying this methodology, we derive fast convergence results for second-order time-evolution systems with dry friction, asymptotically vanishing viscous damping, and implicit Hessian-driven damping.
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Dates et versions

hal-04164463 , version 1 (18-07-2023)

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  • HAL Id : hal-04164463 , version 1

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Samir Adly, Hedy Attouch, Manh Hung Le. A doubly nonlinear evolution system with threshold effects associated with dry friction. 2023. ⟨hal-04164463⟩
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