Auxiliary problem principle and inexact variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function

In view of the minimization of a function which is the sum of a differentiable function $f$ and a convex function $g$ we introduce descent methods which can be viewed as produced by inexact auxiliary problem principleor inexact variable metric forward-backward algorithm. Assuming that the global objective function satisfies the Kurdyka-Lojasiewicz inequalitywe prove the convergence of the proposed algorithm weakening assumptions found in previous works.