Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces
TL;DR: A Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces is proposed and strong convergence theorem for the algorithm is proved.
Abstract: In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.
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TL;DR: This paper presents two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation and proves that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous.
Abstract: In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators to be co-coercive. This restrictive co-coercive assumption has been dispensed with in some methods, many of which require a product space formulation of the problem. However, it has been discovered that this product space formulation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting structure of the split variational inequality problem. In this paper, we present two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation. We prove that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous. Also, we provide several numerical experiments of the proposed methods in comparison with other related methods in the literature.
49 citations
TL;DR: In this paper, an inertial extrapolation method for solving generalized split feasibility problems over the solution set of monotone variational inclusion problems in real Hilbert space is proposed. But this method is not suitable for real Hilbert spaces.
Abstract: In this paper, we propose a new inertial extrapolation method for solving the generalized split feasibility problems over the solution set of monotone variational inclusion problems in real Hilbert...
47 citations
TL;DR: In this paper, an iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize was proposed.
Abstract: In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.
42 citations
TL;DR: In this paper, Tseng's extragradient algorithm with self-adaptive step size was proposed to solve the variational inequality problem (VIP) and the fixed point problem.
Abstract: In this paper, we propose and study new inertial viscosity Tseng's extragradient algorithms with self-adaptive step size to solve the variational inequality problem (VIP) and the fixed point proble...
36 citations
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TL;DR: It is shown that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties, which makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems.
Abstract: We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.
2,645 citations
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28 Sep 1990TL;DR: In this paper, the basic fixed point theorems for non-pansive mappings are discussed and weak sequential approximations are proposed for linear mappings with normal structure and smoothness.
Abstract: Introduction 1. Preliminaries 2. Banach's contraction principle 3. Nonexpansive mappings: introduction 4. The basic fixed point theorems for nonexpansive mappings 5. Scaling the convexity of the unit ball 6. The modulus of convexity and normal structure 7. Normal structure and smoothness 8. Conditions involving compactness 9. Sequential approximation techniques 10. Weak sequential approximations 11. Properties of fixed point sets and minimal sets 12. Special properties of Hilbert space 13. Applications to accretivity 14. Nonstandard methods 15. Set-valued mappings 16. Uniformly Lipschitzian mappings 17. Rotative mappings 18. The theorems of Brouwer and Schauder 19. Lipschitzian mappings 20. Minimal displacement 21. The retraction problem References.
1,466 citations
TL;DR: Using an extension of Pierra's product space formalism, it is shown here that a multiprojection algorithm converges and is fully simultaneous, i.e., it uses in each iterative stepall sets of the convex feasibility problem.
Abstract: Generalized distances give rise to generalized projections into convex sets. An important question is whether or not one can use within the same projection algorithm different types of such generalized projections. This question has practical consequences in the area of signal detection and image recovery in situations that can be formulated mathematically as a convex feasibility problem. Using an extension of Pierra's product space formalism, we show here that a multiprojection algorithm converges. Our algorithm is fully simultaneous, i.e., it uses in each iterative stepall sets of the convex feasibility problem. Different multiprojection algorithms can be derived from our algorithmic scheme by a judicious choice of the Bregman functions which govern the process. As a by-product of our investigation we also obtain blockiterative schemes for certain kinds of linearly constraned optimization problems.
1,085 citations
"Halpern-type iterative process for ..." refers methods in this paper
...(1.1) The first instance of SIP (1.1) is the split feasibility problem (in short, SFP) introduced by Censor and Elfving [13], where X and Y are Euclidean spaces and IP1 and IP2 are convex feasibility problems....
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...1) is the split feasibility problem (in short, SFP) introduced by Censor and Elfving [13], where X and Y are Euclidean spaces and IP1 and IP2 are convex feasibility problems....
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993 citations
TL;DR: In this article, it was shown that for any two monotone operators Tx and T2 from X to X*, the operator F», + T2 is again monotonous.
Abstract: is called the effective domain of F, and F is said to be locally bounded at a point x e D(T) if there exists a neighborhood U of x such that the set (1.4) T(U) = (J{T(u)\ueU} is a bounded subset of X. It is apparent that, given any two monotone operators Tx and T2 from X to X*, the operator F», + T2 is again monotone, where (1 5) (Ti + T2)(x) = Tx(x) + T2(x) = {*? +x% I xf e Tx(x), xt e T2(x)}. If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal—some sort of condition is needed, since for example the graph of Tx + T2 can even be empty (as happens when D(Tx) n D(T2)= 0). The problem of determining conditions under which Tx + T2 is maximal turns out to be of fundamental importance in the theory of monotone operators. Results in this direction have been proved by Lescarret [9] and Browder [5], [6], [7]. The strongest result which is known at present is :
922 citations
"Halpern-type iterative process for ..." refers background in this paper
...An example of a maximal monotone operator is ∂f (x), where f is a proper convex and lower semicontinuous function on E (see [48])....
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...8) Also, we know that if E is a smooth, strictly convex, and reflexive Banach space, then F is maximal monotone if and only if R(J + λF) = E∗ (see [48])....
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...It is known that if F is a maximal monotone operator, then R(I +λJ−1F) = E (see [7, 48])....
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