Backward Penalty Schemes for Monotone Inclusion Problems

Abstract

Abstract. In this paper we are concerned with solving monotone inclusion problems expressed by the sum of a set-valued maximally monotone operator with a single-valued maximally monotone one and the normal cone to the nonempty set of zeros of another set-valued maximally monotone operator. Depending on the nature of the single-valued operator, we will propose two iterative penalty schemes, both addressing the set-valued operators via backward steps. The single-valued operator will be evaluated via a single forward step if it is cocoercive, and via two forward steps if it is monotone and Lipschitz continuous. The latter situation represents the starting point for dealing with complexly structured monotone inclusion problems from algorithmic point of view.

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

“…We present a situation where Assumption 2(III) holds and refer to [10] for further examples. For instance, if…”

“…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.…”

Generalized forward–backward splitting with penalization for monotone inclusion problems

“…(say, A := A 1 ). They proposed a forward-backward algorithm of penalty type for solving problem (6). In each iteration, the algorithm performs a forward step with the operator B together with the penalization term with respect to C and a backward step by the resolvent of A, that is,…”

“…To guarantee the convergence of the sequence generated by their proposed algorithm, they introduced a condition formulated by using the Fitzpatrick function associated with the operator C (see Assumption 3.2 (H2)). In their paper, they also presented the forward-backward-forward algorithm, which is known as Tseng's type algorithm with penalty term, for solving (6) in the case where B and C are monotone and Lipschitz continuous. As a continuation of these developments, in [9], the same authors focused on solving a monotone inclusion problem involving linearly composed and parallel-sum monotone operators and the normal cone to the set of zeros of another monotone and Lipschitz continuous operator.…”

“…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. On the other hand, Passty [26] considered a backward splitting method for solving MIP in the case where B = C = 0.…”