generalized epsilon-descent algorithm

Given the problem of minimizing a possibly nonconvex and nonsmooth function in a real Hilbert space, we present a generalized epsilon-descent algorithm motivated from the descent method introduced by Attouch et al. [Math. Program. 137 (2013) 91–129] with two essential additions, we consider scalar errors on the sufficient descent condition, as well as, on the relative inexact optimality condition. Under general conditions on the function to be minimized, we obtain that all accumulation points of the sequences generated by the algorithm, if they exist, are generalized critical limit points of the objective function.