Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces

By Genaro López, Victoria Martín-Márquez, Fenghui Wang, Hong-Kun Xu. 2012

157 citations recorded by the source corpus.

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.

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

View Paper: https://www.semanticscholar.org/paper/1ca906bf2401ef98bf912482cf6e833dac510b80

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In Computational Mechanics

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Citations in the corpus

A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, methodology): “1) is the forward–backward splitting method [6,27,34,36,48,49,53,56,61] which is defined by the following manner: x1 ∈ H and xn+1 = (I + rB)−1(xn − rAxn), n ≥ 1, (1.”
An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces (2018, unknown): “1), the forward–backward splitting method [11,18,28,29,33,41] is usually employed and is defined by the following manner: x1 ∈ H and xn+1 = (I + rB)−1(xn − rAxn), n ≥ 1, (1.”
Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces (2018, background): “6 [12]: Let X be a uniformly convex and q-uniformly smooth Banach space with q ∈ (1, 2].”