Proximal Activation of Smooth Functions in Splitting Algorithms for Convex Image Recovery

A strongly convergent Krasnosel'ski\vı-Mann-type algorithm for finding a common fixed point of a countably infinite family of nonexpansive operators in Hilbert spaces

“…denotes the proximal operator of f . For a nonempty closed and convex set C ⊆ H, one has that P C = prox δ C , where The strongly convergent forward-backward algorithm with variable step sizes for solving (12) that we propose in this section will be formulated as a particular case of (10) for R n = J γnA (Id −γ n B), for n ≥ 0, where inf n≥0 γ n > 0. To this end we will prove that (R n ) n≥0 fulfills the asymptotic condition (11) (see [14,Corollary 17]).…”

“…f is identical zero and g is the function in the objective); (2) to divide the objective into two parts and evaluate one of the two smooth functions via its proximal operator, hence, end up with a proximal-gradient scheme. We pursued both approaches, by taking also into account [12], which suggests that the evaluation of a smooth objective via its proximal operator may be advantageous in terms of computational performance compared to evaluating the whole objective through its gradient.…”