Forward-Backward and Tseng's Type Penalty Schemes for Monotone Inclusion Problems

TL;DR

This paper introduces new forward-backward and forward-backward-forward algorithms for solving complex monotone inclusion problems in Hilbert spaces, extending existing methods to handle more general structures with convergence guarantees.

Contribution

It proposes novel penalty schemes for monotone inclusions, extending prior algorithms to include Lipschitz conditions and compositions, with convergence analysis based on Fitzpatrick functions.

Findings

The algorithms converge weakly under specified conditions.

The methods handle more complex problem structures.

Convergence is guaranteed using Fitzpatrick function criteria.

Abstract

We deal with monotone inclusion problems of the form $0\in Ax+Dx+N_C(x)$ in real Hilbert spaces, where $A$ is a maximally monotone operator, $D$ a cocoercive operator and $C$ the nonempty set of zeros of another cocoercive operator. We propose a forward-backward penalty algorithm for solving this problem which extends the one proposed by H. Attouch, M.-O. Czarnecki and J. Peypouquet in [3]. The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set $C$. In the second part we introduce a forward-backward-forward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.