A proximal iterative approach to a non-convex optimization problem

We consider a variable Krasnosel’skii-Mann algorithm for approximating critical points of a prox-regular function or equivalently for finding fixed-points of its proximal mapping prox_λf. The novelty of our approach is that the latter is not non-expansive any longer. We prove that the sequence generated by such algorithm (via the formula x_k+1=(1−α_k)x_k+α_kprox_λkfx_k, where (α_k) is a sequence in (0,1)), is an approximate fixed-point of the proximal mapping and converges provided that the function under consideration satisfies a local metric regularity condition.