Dynamical systems and forward–backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator

Abstract

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators , where is the subdifferential of a convex lower semicontinuous function , and is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials which was considered in Abbas, Attouch, Svaiter JOTA, 2014. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by , and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems, thereby extending the known results. The time discretization of these dynamics gives various forward–backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multicriteria decision processes.

Keywords:

• structured monotone inclusions
• forward–backward algorithms
• subdifferential operators
• cocoercive operators
• proximal-gradient method
• dissipative dynamics
• Lyapunov analysis
• weak asymptotic convergence
• Levenberg-Marquardt regularization
• multiobjective decision

AMS Subject Classifications:

• 34G25
• 47J25
• 47J30
• 47J35
• 49M15
• 49M37
• 65K15
• 90C25
• 90C53

Notes

1 Indeed, the result is true under the more general assumption, uniformly convex. We thank the referee for pointing this result.