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First order inertial optimization algorithms with threshold effects associated with dry friction

Abstract : In a Hilbert space setting, we consider a new first order optimization algorithm which is obtained by temporal discretization of a damped inertial dynamic involving dry friction. The function f to be minimized is assumed to be differentiable (not necessarily convex). The dry friction potential function φ, which has a sharp minimum at the origin, enters the algorithm via its proximal mapping, which acts as a soft thresholding operator on the sum of the velocity and the gradient terms. After a finite number of steps, the structure of the algorithm changes, losing its inertial character to become the steepest descent method. The geometric damping driven by the Hessian of f makes it possible to control and attenuate the oscillations. The algorithm generates convergent sequences when f is convex, and in the nonconvex case when f satisfies the Kurdyka-Lojasiewicz property. As a remarkable property, the convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence towards zero. The study is then extended to the case of a nonsmooth convex function f , in which case the algorithm involves the proximal operators of f and φ separately. Then, applications are given to the Lasso problem and nonsmooth d.c. programming.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03284220
Contributor : Samir Adly <>
Submitted on : Monday, July 12, 2021 - 2:14:32 PM
Last modification on : Wednesday, September 15, 2021 - 12:08:03 PM

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  • HAL Id : hal-03284220, version 1

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Samir Adly, Hedy Attouch, Manh Hung Le. First order inertial optimization algorithms with threshold effects associated with dry friction. 2021. ⟨hal-03284220⟩

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