Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

Abstract

In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

Semi-Decentralized Generalized Nash Equilibrium Seeking in Monotone Aggregative Games

“…with Φ as in (22). The iteration in (34) corresponds to the inertial proximal-point method in [44] applied to the mapping T preconditioned with Φ C . When T is maximally monotone (which follows by Assumption 6) and the step sizes in the main diagonal of Φ are set such that Φ C 0, then J Φ −1 C T is firmly nonexpansive ( 12 −averaged) w.r.t.…”

“…the Φ C −induced norm, i.e., • Φ C . Moreover, if the inertial parameter θ k is nondecreasing and small enough, then the inertial fixed-point iteration (34) converges to some…”

“…By combining (66) and (67), we obtain the fixed-point iteration in (34), that corresponds to the iteration in [44, Th. 2.1] applied on Φ −1 C T and, thus, concludes the proof.…”

“…The fixed-point iteration (34), that corresponds to Algorithm 6 by the first part of this proof, is the inertial proximal-point algorithm applied on the operator Φ −1 C T . The convergence of (34) to some ω * = col(x * , λ * ) ∈ zer(Φ −1 C T ) follows by [44, Th.…”

On the convergence of an inertial proximal algorithm with a Tikhonov regularization term

“…where we denote α k = 1 − α k q and c k = c k p . For a better insight in inertial proximal algorithms we refer to [13,14,16,19,24,26,27,28] Note that c k x k is a Tikhonov regularization term, which may assure the strong convergence of a generated sequence to the minimizer of minimal norm of the objective function f , (see [1,4,6,9,10,11,15,18,20,23,25,31,32]).…”