Abstract
In a Hilbert space H, we introduce a new class of proximal-gradient algorithms with finite convergence properties. These algorithms naturally occur as discrete temporal versions of an inertial differential inclusion which is damped under the joint action of three dampings: a viscous damping, a geometric damping driven by the Hessian and a dry friction damping. The function f : H → R to be minimized is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient. The dry friction damping function φ : H → R + is convex with a sharp minimum at the origin, (typically φ(x) = r x with r > 0). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. The geometrical damping driven by the Hessian intervenes in the dynamics in the form ∇ 2 f (x(t))ẋ(t). By treating this term as the time derivative of ∇f (x(t)), this gives, in discretized form, first-order algorithms. The Hessian driven damping allows to control and to attenuate the oscillations which occur naturally with the inertial effect. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero. Then, replacing the potential function f by its Moreau envelope, we extend the results to the case of a nonsmooth convex function f. In this case, the algorithm involves the proximal operators of f and φ separately. Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension in the case of additive composite optimization, thus leading to splitting methods. Numerical experiments are given for Lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of two variants of the Nesterov accelerated method with dry friction and Hessian-driven viscous damping. Mathematics Subject Classifications: 37N40, 34A60, 34G25, 49K24, 70F40.