  This paper presents and investigates an inexact proximal gradient method for
solving composite convex optimization problems characterized by an objective
function composed of a sum of a full domain differentiable convex function and
a non-differentiable convex function. We introduce an explicit linesearch
strategy that requires only a relative inexact solution of the proximal
subproblem per iteration. We prove the convergence of the sequence generated by
our scheme and establish its iteration complexity, considering both the
functional values and a residual associated with first-order stationary
solutions. Additionally, we provide numerical experiments to illustrate the
practical efficacy of our method.
