A generalized forward-backward splitting (Q2873234)

An operator \(B\) on a real Hilbert space \(\mathcal H\) is called cocercive if there exists \(\alpha >0\) such that \(\langle B(x)-B(y), x-y \rangle \geq \alpha {\|B(x)-B(y)\|}^2\) for all \(x,y \in \mathcal H\). A set-valued map \(A:\mathcal H \rightarrow \)\(2^{\mathcal H}\) is called monotone if \(\langle x-y,u-v \rangle \geq 0\) for all \(x,y \in \mathcal H\) and for every \(u \in Ax\), \(v \in Ay\). \(A\) is called maximal monotone if it is maximal with respect to graph inclusions. The authors study the following monotone inclusion problem: Find \(x \in \mathcal H\) such that \(x \in B(x) + \sum_{i=1}^n A_i(x)\), for maximal monotone maps \(A_i\) and for a fixed positive integer \(n\). They propose a generalized forward-backward splitting algorithm and prove its convergence. Applications are illustrated for problems in imaging.