An algorithmic scheme for solving, over any real Hilbert space ℋ, monotone inclusion problems of the form

find *x* ∈ zer { ∑_{i=1}^{n} *A _{i}* +

- all considered operators are maximally monotone,
*B*is cocoercive,- for all
*i*∈ {1,…,n},*A*is simple, in the sense that we can compute easily its resolvent:_{i}*J*:= (Id +_{Ai}*A*)_{i}^{-1}.

In particular, it enables minimization over ℋ of convex problems of the form

find *x*∈ argmin { ∑_{i=1}^{n} *g _{i}* +

- all considered functions are lower semicontinuous, proper, and convex from ℋ to ]−∞,+∞],
*f*is differentiable with Lipschitz-continuous gradient,- for all
*i*∈ {1,…,n},*g*is simple, in the sense that we can compute easily its proximity operator: prox_{i}_{gi}:*x*↦ argmin_{y∈ℋ}{

||1 2 *x*−*y*||^{2}+*g*(_{i}*y*) } .

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Download MATLAB codes and materials used in this paper.

Some supplementary results: deblurring task - inpainting task - composite task - composite task with TV regularization.

A more complete version of this work can be found in chapters III and IV of my Ph.D. thesis.

See also the preconditioned version.