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*) .

Published in

Source code of numerical simulations used in the paper is not maintained anymore, feel free to contact me for any question.

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, or the forward-Douglas–Rachford version.