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Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions

Radu Ioan Bot (radu.bot***at***univie.ac.at)
Ernö Robert Csetnek (ernoe.robert.csetnek***at***univie.ac.at)

Abstract: We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to a zero of the sum, provided the latter is strongly monotone. We derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper, convex and lower semicontinuous function with a smooth convex one provided the objective function fulfills a strong convexity assumption. In the particular case of minimizing a smooth and strongly convex function, we prove that its values converge along the trajectory to its minimum value with exponential rate, too.

Keywords: dynamical systems, strongly monotone inclusions, continuous forward-backward method, convergence rates, convex optimization problems

Category 1: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

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Entry Submitted: 04/08/2015
Entry Accepted: 04/08/2015
Entry Last Modified: 04/09/2015

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