Abstract: In this talk we present an abstract forward-backward splitting algorithm that can be interpreted in terms of a generalized convex-concave procedure. The algorithm unifies many popular existing methods such as the classical convex-concave procedure, Bregman and classical forward-backward splitting, proximal point algorithms with \phi-divergences, natural gradient descent and motivates entirely new algorithms such as an anisotropic generalization of forward-backward splitting. We analyze the algorithms’ sublinear and linear convergence under an abstract proximal PL-inequality. In the spirit of the classical convex-concave procedure we also study certain equivalent reformulations of the optimization problem involving generalized conjugate functions obtained via a double-min duality. This viewpoint leads to a natural generalisation of the Moreau and forward-backward envelope (called c-conjugates in the context of generalised concavity) that serve as real-valued continuous surrogates preserving (local) minima and stationary points. Leveraging the generalized implicit function theorem we study the 1st and 2nd order-differentiability properties of these envelopes. Ultimately this is used to obtain globalized (quasi-)Newton accelerated versions of the aforementioned algorithms.