Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems

Abstract

We deal with monotone inclusion problems of the form 0 ∈ Ax + Dx + N C (x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forwardbackward penalty algorithm for solving this problem which extends the one proposed by H. Attouch, M.-O. Czarnecki and J. Peypouquet in [3]. The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set C. In the second part we introduce a forward-backwardforward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.

Generalized forward–backward splitting with penalization for monotone inclusion problems

“…Remark 3.1 (i) When m = 1, GFBP reduces to the algorithm proposed and investigated in [10] for solving (6). On the other hand, if B(x) = C(x) = 0 for all x ∈ H, then GFBP turns out to be the m−fold backward algorithm proposed in [26].…”

“…In order to approximate a solution of MIP, we remark that Boţ and Csetnek [10] investigated a particular situation of MIP of the form…”

“…An application on image inpainting has been also presented. In [12], the same authors considered the penalty type algorithm with inertial effect of the one proposed in [10] for solving (6). In a similar fashion, Banert and Boţ [6] proposed the backward penalty type algorithm for solving (6) in the case where C is a maximally monotone operator.…”

An Inertial Proximal-Gradient Penalization Scheme for Constrained Convex Optimization Problems

“…where for the last inequality we use (17) and (18). Since λ n → 0 as n → +∞, there exists N 1 ∈ N, such that for every n ≥ N 1 , we have…”

Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

“…This represented the starting point for the design and development of numerical algorithms for solving the minimization problem (1), several variants of it involving also nonsmooth data up to monotone inclusions that are related to optimality systems of constrained optimization problems. We refer the reader to [4][5][6][7][8]10,11,[13][14][15][20][21][22][23]33,35] and the references therein for more insights into this research topic.…”