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:

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:

Published in SIAM Journal of Imaging Sciences:
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Some supplementary results: deblurring task - inpainting task - composite task - composite task with TV regularization