# Generalized Forward-Backward Splitting

It solves, over any real Hilbert space $\mathcal{H}$, monotone inclusion problems of the form
$\text{Find } x \in \left\{ \text{zer} \left( B+\sum_{i=1}^{n} A_i \right) \stackrel{\mathrm{\scriptscriptstyle{def}}}{=} \{ x \in \mathcal{H} \mid 0 \in B x + \sum_{i=1}^{n} A_i x \} \right\}~,$ where:

• all considered operators are maximal monotone,
• $B$ is cocoercive,
• all the $A_i$'s are simple in the sense that we can compute easily their resolvant: $J_{A_i} \stackrel{\mathrm{\scriptscriptstyle{def}}}{=} \left( \text{Id}+A_i \right)^{-1}$.

In particular, it enables minimization over $\mathcal{H}$ of convex problems of the form
$\min_{x \in \mathcal{H}} \left\{ F(x) \stackrel{\mathrm{\scriptscriptstyle{def}}}{=} f(x) + \sum_{i=1}^{n} g_i(x) \right\}~,$ where:

• all considered functions belong to $\Gamma_0(\mathcal{H})$, the class of lower semi-continuous, proper, convex functions from $\mathcal{H}$ to $]{-}\infty,{+}\infty]$,
• $f$ is differentiable with Lipschitz-continuous gradient,
• all the $g_i$'s are simple in the sense that we can compute easily their Moreau's proximity operator: $\text{prox}_{g_i}(x) \stackrel{\mathrm{\scriptscriptstyle{def}}}{=} \text{argmin}_{y} \frac{1}{2} {\Vert x - y \Vert}^2 + g_i(y)$.

Published in SIAM Journal of Imaging Sciences:
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