Nonlinear Forward-Backward Splitting with Projection Correction

We propose and analyze a versatile and general algorithm called nonlinear forward-backward splitting (NOFOB). The algorithm consists of two steps; first an evaluation of a nonlinear forward-backward map followed by a relaxed projection onto the separating hyperplane it constructs. The key of the method is the nonlinearity in the forward-backward step, where the backward part is based on a nonlinear resolvent construction that allows for the kernel in the resolvent to be a nonlinear single-valued maximal monotone operator. This generalizes the standard resolvent as well as the Bregman resolvent, whose resolvent kernels are gradients of convex functions. This construction opens up for a new understanding of many existing operator splitting methods and paves the way for devising new algorithms. In particular, we present a four-operator splitting method as a special case of NOFOB that relies nonlinearity and nonsymmetry in the forward-backward kernel. We show that forward-backward-forward splitting (FBF), forward-backward-half-forward splitting (FBHF), asymmetric forward-backward-adjoint splitting (AFBA) with its many special cases, as well as synchronous projective splitting are special cases of the four-operator splitting method and hence of NOFOB. We also show that standard formulations of FB(H)F use smaller relaxations in the projections than allowed in NOFOB. Besides proving convergence for NOFOB, we show linear convergence under a metric subregularity assumption, which in a unified manner shows (in some cases new) linear convergence results for its special cases.