Warped proximal iterations for monotone inclusions

Nonlinear Forward-Backward Splitting with Momentum Correction

“…By a primal-dual method, we mean a method that, instead of solving (10) directly, solves the equivalent primal-dual problem of finding y ∈ K and z ∈ G such that…”

“…Compared to [10,11], the elements of the sequence (x k ) k∈N are given directly by a nonlinear forward-backward step and do not need an extra projection step. Convergence is instead ensured by the addition of the corrective term u k to the forward step.…”

“…Nonlinear resolvents -or warped resolvents in the terminology of [10] -were recently explored in [10,11] with precursors available in [12,13]. These works are preceded by, or developed in parallel with, several other generalizations to the concept of a resolvent.…”

“…However, this extra freedom may come at a cost. The algorithms of [10,11,13,14] all need an extra corrective projection step to ensure that any nonlinearities and asymmetries of the kernel do not prevent convergence. The primary contribution of this paper is a different approach for correcting the update, removing the need to perform a potentially expensive projection.…”

“…The strength of nonlinear resolvents lies in their substantial modeling power which allows for a unified view of a large set of algorithms. Both [10,11] present numerous algorithms that can be interpreted as forward-backward methods with nonlinear resolvents. Our new nonlinear forward-backward method further expands on these modeling capabilities and the second half of this paper is dedicated to deriving both new and existing algorithms as special cases.…”