Second-order dynamical systems with penalty terms associated to monotone inclusions

Abstract

In this paper we investigate in a Hilbert space setting a second order dynamical system of the formẍwhere A : H ⇒ H is a maximal monotone operator, J λ(t)A : H −→ H is the resolvent operator of λ(t)A and D, B : H → H are cocoercive operators, and λ, β : [0, +∞) → (0, +∞), and γ : [0, +∞) → (0, +∞) are step size, penalization and, respectively, damping functions, all depending on time. We show the existence and uniqueness of strong global solutions in the framework of the Cauchy-Lipschitz-Picard Theorem and prove ergodic asymptotic convergence for the generated trajectories to a zero of the operator A + D + N C , where C = zer B and N C denotes the normal cone operator of C. To this end we use Lyapunov analysis combined with the celebrated Opial Lemma in its ergodic continuous version. Furthermore, we show strong convergence for trajectories to the unique zero of A + D + N C , provided that A is a strongly monotone operator.

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“…Another interesting future direction is to extend the stability results to second order in-time dynamical systems and algorithms, which may provide the same stability results with earlier stopping times (and thus less computational resources) (see e.g. [1], [6]). Finally, a future line of research is related with the study of regularization properties of the continuous flow associated to the primaldual algorithm [12], [19].…”