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 proxλfproxλf. 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 xk+1=(1−αk)xk+αkproxλkfxkxk+1=(1−αk)xk+αkproxλkfxk, where (αk)(αk) is a sequence in (0,1)(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.