A Generalized Forward-Backward Splitting

An algorithmic scheme for solving, over any real Hilbert space ℋ, monotone inclusion problems of the form
find x ∈ zer (∑_i=1ⁿ A_i + B ) , where:

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

In particular, it enables minimization over ℋ of convex problems of the form
find x∈ argmin ∑_i=1ⁿ g_i + f , where:

• 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_i is simple, in the sense that we can compute easily its proximity operator: prox_gi: x ↦ argmin_y∈ℋ  

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  ||x − y||² + g_i(y) .