Three new iterative methods for solving inclusion problems and related problems
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Three new iterative methods for solving inclusion problems and related problems Aviv Gibali1,2
· Duong Viet Thong3 · Nguyen The Vinh4
Received: 7 December 2019 / Revised: 5 January 2020 / Accepted: 1 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we study the variational inclusion problem which consists of finding zeros of the sum of a single and multivalued mappings in real Hilbert spaces. Motivated by the viscosity approximation, projection and contraction and inertial forward–backward splitting methods, we introduce two new forward–backward splitting methods for solving this variational inclusion. We present weak and strong convergence theorems for the proposed methods under suitable conditions. Our work generalize and extend some related results in the literature. Several numerical examples illustrate the potential applicability of the methods and comparisons with related methods emphasize it further. Keywords Projection and contraction method · Inertial forward–backward splitting method · Viscosity method · Zero point Mathematics Subject Classification 65Y05 · 65K15 · 68W10 · 47H09 · 47J25
Communicated by José R Fernández.
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Duong Viet Thong [email protected] Aviv Gibali [email protected] Nguyen The Vinh [email protected]
1
Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel
2
The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838 Haifa, Israel
3
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
4
Department of Mathematics, University of Transport and Communications, Hanoi, Vietnam 0123456789().: V,-vol
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A. Gibali et al.
1 Introduction In this paper, we study the following inclusion problem: find x ∗ ∈ H such that 0 ∈ Ax ∗ + Bx ∗ ,
(1)
where H is a real Hilbert space, A : H → H is a monotone operator and B : H → 2 H is a maximal monotone operator. The solution set of (1) is denoted by (A + B)−1 (0). This problem plays an important role in many fields, such as equilibrium problems, fixed point problems, variational inequalities, and composite minimization problems, see, for example, (Aoyama et al. 2008; Brézis 1973; Bauschke and Combettes 2011; Zeidler 1990). To be more precise, many problems in signal processing, computer vision and machine learning can be modeled mathematically as this formulation, see (Beck and Teboulle 2009; Combettes and Wajs 2005; Raguet et al. 2013; Verma et al. 2018) and the references therein. For solving the problem (1), the so-called forward–backward splitting method is given as follows: x0 ∈ H , (2) xn+1 = (I + λn B)−1 (I − λn A)xn , where λn > 0. The forward–backward splitting algorithm for monotone inclusion problems was first introduced by Lions and Mercier (1979). In the work of Lions and Mercier, other splitting methods, such as Peaceman–Rachford algorithm (Peaceman and Rachford 1955) and Douglas–Rachford (Douglas and Rachford 19
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