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

Samir Adly (), Hedy Attouch () and Manh Hung Le ()
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Samir Adly: Université de Limoges
Hedy Attouch: Université Montpellier, CNRS, Place Eugène Bataillon
Manh Hung Le: Université de Limoges

Computational Optimization and Applications, 2023, vol. 86, issue 3, No 2, 843 pages

Abstract: Abstract In a Hilbert space setting, we consider new first order optimization algorithms which are obtained by temporal discretization of a damped inertial autonomous dynamic involving dry friction. The function f to be minimized is assumed to be differentiable (not necessarily convex). The dry friction potential function $$ \varphi $$ 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. The convergence results are robust with respect to numerical errors, and perturbations. 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 $$\varphi $$ φ separately. Applications are given to the Lasso problem and nonsmooth d.c. programming.

Keywords: Proximal-gradient algorithms; Inertial methods; Dry friction; Hessian-driven damping; Soft thresholding; Kurdyka–Lojasiewicz property; Lasso problem; D.c. optimization; Errors; 37N40; 34A60; 34G25; 49K24; 70F40 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00509-9

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