About DC and split composite minimization problems

The aim of this paper is twofold.Firstly, to prove the linear convergence of a linearized proximal algorithm for solving DC composite optimization problems under suitable assumptions.In the DC optimization setting, this sharpens recent results.Secondly, to sketch out some ideas for solving split convex composite minimization problems by means of an alternative formulation based on a generalization of an infimal post-composition approach developed very recently.This will most probably rely on a conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space even in infinite dimensional spaces together very possibly with new notions.The goal is to move from an approach based on the composition of a convex function by a linear operator to a composition by a suitable map.