Nonlinear Forward-Backward Splitting with Projection Correction

Understand

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.