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

Radu Ioan Bot(radu.bot***at***mathematik.tu-chemnitz.de)
Ernö Robert Csetnek(robert.csetnek***at***mathematik.tu-chemnitz.de)

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.

Keywords: maximally monotone operator, Fitzpatrick function, resolvent, cocoercive operator, Lipschitz continuous operator, forward-backward algorithm, forward-backward-forward algorithm, subdifferential, Fenchel conjugate

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

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Entry Submitted: 06/03/2013
Entry Accepted: 06/03/2013
Entry Last Modified: 06/03/2013

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