Forward-backward splitting under the light of generalized convexity

In this paper we present a unifying framework for continuous optimization methods grounded in the concept of generalized convexity. Utilizing the powerful theory of $\Phi$-convexity, we propose a conceptual algorithm that extends the classical difference-of-convex method, encompassing a broad spectrum of optimization algorithms. Relying exclusively on the tools of generalized convexity we develop a gap function analysis that strictly characterizes the decrease of the function values, leading to simplified and unified convergence results. As an outcome of this analysis, we naturally obtain a generalized PL inequality which ensures $q$-linear convergence rates of the proposed method, incorporating various well-established conditions from the existing literature. Moreover we propose a $\Phi$-Bregman proximal point interpretation of the scheme that allows us to capture conditions that lead to sublinear rates under convexity.