“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem

In this paper, we focus on studying the split feasibility problem (SFP), which has many applications in signal processing and image reconstruction. A popular technique is to employ the iterative method which is so called the relaxed CQ algorithm. However, the speed of convergence usually depends on the way of selecting the step size of such algorithms. We aim to suggest a new hybrid CQ algorithm for the SFP by using the self adaptive and the line-search techniques. There is no computation on the inverse and the spectral radius of a matrix. We then prove the weak convergence theorem under mild conditions. Numerical experiments are included to illustrate its performance in compressed sensing. Some comparisons are also given to show the efficiency with other CQ methods in the literature.

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“…Recently, many authors establish weak and strong convergence theorems for the SFP (see also [10,11]).…”

Section: Introductionmentioning

confidence: 99%

A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

2019

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“…Moreover, many authors have studied the methods and algorithms to approximate a solution of the problem (1), for instance, descent algorithms in Konnov and Ali [5], Konnov and Pinyagina [6], Charitha [7], and Lorenzo et al [8]. For more details, refer to Ceng [9], Yao et al [10][11][12], Qin et al [13,14], Hung and Muu [15], Quoc et al [16], Santos et al [17], Thuy et al [18], Rockafellar [19], Moudafi [20,21], Muu and Otelli [22], and Dong et al [23].…”

Section: Introductionmentioning

confidence: 99%

Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

Tang

Cho

2019

Mathematics

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In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian and Fakhri, a new self-adaptive step size algorithm is proposed to find a common element of the solution set of the problems SVIP and EP. Convergence theorems are established under suitable conditions for the algorithm and application to the common solution of the fixed point problem, and the split convex optimization problem is considered. Finally, the performances and computational experiments are presented and a comparison with the related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

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“…It has been found that the SFP can also be used in the air traffic flow management problems. Many researchers studied the SFP and introduced various algorithms to solve it (see [5][6][7][8][9][10][11][12][13][14][15] and references therein). The original algorithm introduced in [1] involves the computation of the inverse A −1 (assuming the existence of the inverse of A) and thus does not become popular.…”

Section: Introductionmentioning

confidence: 99%

Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

2019

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The split feasibility problem models inverse problems arising from phase retrievals problems and intensity-modulated radiation therapy. For solving the split feasibility problem, Xu proposed a relaxed CQ algorithm that only involves projections onto half-spaces. In this paper, we use the dual variable to propose a new relaxed CQ iterative algorithm that generalizes Xu’s relaxed CQ algorithm in real Hilbert spaces. By using projections onto half-spaces instead of those onto closed convex sets, the proposed algorithm is implementable. Moreover, we present modified relaxed CQ algorithm with viscosity approximation method. Under suitable conditions, global weak and strong convergence of the proposed algorithms are proved. Some numerical experiments are also presented to illustrate the effectiveness of the proposed algorithms. Our results improve and extend the corresponding results of Xu and some others.

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“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem

Cited by 50 publications

References 37 publications

A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

Modified Relaxed CQ Iterative Algorithms for the Split Feasibility Problem

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