Inertial Douglas–Rachford splitting for monotone inclusion problems

Abstract

Abstract. We propose an inertial Douglas-Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigate its convergence properties. To this end we formulate first the inertial version of the Krasnosel'skiȋ-Mann algorithm for approximating the set of fixed points of a nonexpansive operator, for which we also provide an exhaustive convergence analysis. By using a product space approach we employ these results to the solving of monotone inclusion problems involving linearly composed and parallel-sum type operators and provide in this way iterative schemes where each of the maximally monotone mappings is accessed separately via its resolvent. We consider also the special instance of solving a primal-dual pair of nonsmooth convex optimization problems and illustrate the theoretical results via some numerical experiments in clustering and location theory.

Quasi-Nonexpansive Iterations on the Affine Hull of Orbits: From Mann's Mean Value Algorithm to Inertial Methods

“…Then T x n − x n → 0 and there exists x ∈ Fix T such that x n ⇀ x. In the case when T is nonexpansive, this result appears in [14,Theorem 5].…”

“…Proof. We use arguments similar to those used in [3,14]. It follows from (c) that (∀n ∈ N) 0 < ϑ/φ n − η 2 ω n+1 ϑ − η 2 (1 + η) − ησ.…”

“…x n = (1 + η n )x n − η n x n−1 η n ∈ [0, 1[ γ n ∈ ]0, +∞[ , (1.6) and where prox f is the proximity operator of f [10,36]. The inertial proximal point algorithm (1.6) has been extended in various directions, e.g., [3,14,17]; see also [5] for further motivation in the context of nonconvex minimization problems.…”

Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions

“…where A and B are maximal monotone and B is cocoercive 1 . Other special cases were introduced as inertial versions of the KM iterations for finding fixed points [33,11]. In this paper we focus on convex optimization, which allows us to obtain less stringent convergence criteria than in those previous studies because we can use properties unique to convex functions.…”

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

“…The resulting iterative schemes share the feature that the next iterate is defined by means of the last two iterates, a fact which induces the inertial effect in the algorithm. Since the works [1,3], one can notice an increasing number of research efforts dedicated to algorithms of inertial type (see [1][2][3]9,[16][17][18][19][24][25][26][27][28][30][31][32]34]). …”