Inertial Douglas-Rachford splitting for monotone inclusion problems

By Radu Ioan Bot, Ernö Robert Csetnek, Christopher Hendrich. Applied Mathematics and Computation, 2014

322 citations recorded by the source corpus.

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'skii?-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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/87823e8ab5c01a3f9f644cd252d9d96d53a015ab

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Citations in the corpus

A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces (2019, background): “In 2015, Bot and Csetnek [39] removed the condition (A2) and substituted (A1) and (A3) with the following conditions: (B1) for each n ≥ 1, {θn} ⊂ [0, α) is nondecreasing with θ1 = 0 and 0 ≤ α < 1, (B2) for each n ≥ 1, δ > α 2(1 + α) + ασ 1 − α2 , 0 ≤ λ ≤ λn ≤ δ − α[α(1 + α) + αδ + σ] δ[1 + α(1 + α) ”
Convergence results of two-step inertial proximal point algorithm (2022, methodology): “The inertial PPA (8) has been adapted to studying inertial Douglas-Rachford splitting method [7,14,17,26,27,37], inertial alternating method of multipliers (ADMM) [15,24,26,31,32] and demonstrated their performance numerically on some imaging and data analysis problems.”
A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, unknown): “Recently, there have been increasing interests in studying inertial type algorithms, see, for example, inertial forward–backward splitting methods [36,46], inertial Douglas–Rachford splitting method [13], inertial ADMM [14,22], and inertial forward–backward–forward method [15].”
A note on the inertial proximal point method (2015, methodology): “, inertial forward-backward splitting methods [28, 27, 6], inertial Douglas-Rachford splitting method [9], inertial ADMM [7], and inertial forward-backward-forward method [8].”
SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD (2018, methodology): “In 2014, they analyzed with Csetnek an inertial DR algorithm and used it to solve clustering problems [53].”