A note on the spectral gradient projection method for nonlinear monotone equations with applications

Abubakar, Auwal Bala; Mohammad, Hassan

doi:10.1007/s40314-020-01151-5

Cited by 47 publications

(42 citation statements)

References 33 publications

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“…We notice that modified spectral gradient algorithms were proposed in [33,34] for solving problem (1), but our algorithm is very different from theirs in the following aspects. Firstly, we give a changeable choice for the inexact parameter in the spectral method.…”

Section: Introductionmentioning

confidence: 89%

A Family of Modified Spectral Projection Methods for Nonlinear Monotone Equations with Convex Constraint

Lin

2020

Mathematical Problems in Engineering

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In this paper, we propose a family of modified spectral projection methods for nonlinear monotone equations with convex constraints, where the spectral parameter is mainly determined by a convex combination of the modified long Barzilai–Borwein stepsize and the modified short Barzilai–Borwein stepsize. We obtain a trial point by the spectral method and then get the iteration point by the projection technique. The algorithm can generate a bounded iterative sequence automatically, and we obtain the global convergence of the proposed method in the sense that every limit point is a solution of the nonlinear equation. The proposed method can be used to resolve the large-scale nonlinear monotone equations with convex constraints including smooth and nonsmooth equations. Numerical results for nonlinear equation problems and the ℓ 1 -norm regularization problem in compressive sensing demonstrate the efficiency and efficacy of our method.

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Section: Introductionmentioning

confidence: 89%

A Family of Modified Spectral Projection Methods for Nonlinear Monotone Equations with Convex Constraint

Lin

2020

Mathematical Problems in Engineering

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“…The parameters selected to implement the algorithm are: τ = 1; ς = 10 −4 ; = 0.55; λ t = 1 (2t+5) 2 ; ζ = 1; µ = 1. In addition, we compare its performance with the iterative shrinkage/thresholding (IST) [58] designed for waveletbased image deconvolution and the PSGM method [59] to reflect the performance of the DF-PRPMHS method in restoring the blurred and noisy images. It is worth noting that the iterative procedure for all the algorithms starts using the same initial point and ends when the tolerance, T ol < 10 −5 .…”

Section: Application To Image Restoration Problemsmentioning

confidence: 99%

A Family of Derivative-Free Conjugate Gradient Methods for Constrained Nonlinear Equations and Image Restoration

Ibrahim

Kumam

2020

IEEE Access

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In this paper, a derivative-free conjugate gradient method for solving nonlinear equations with convex constraints is proposed. The proposed method can be viewed as an extension of the three-term modified Polak-Ribiére-Polyak method (TTPRP) and the three-term Hestenes-Stiefel conjugate gradient method (TTHS) using the projection technique of Solodov and Svaiter [Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, 1998, 355-369]. The proposed method adopts the adaptive line search scheme proposed by Ou and Li [Journal of Applied Mathematics and Computing 56.1-2 (2018): 195-216] which reduces the computational cost of the method. Under the assumption that the underlying operator is Lipschitz continuous and satisfies a weaker condition of monotonicity, the global convergence of the proposed method is established. Furthermore, the proposed method is extended to solve image restoration problem arising in compressive sensing. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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“…Reconstruction of sparse signal in compressive sensing is shown in this section to illustrate the performance of Algorithm 2.1. We compared Algorithm 2.1 to the following methods in the literature to demonstrate the efficiency of our proposed method in signal reconstruction: SGCS (Xiao et al, 2011), CGD (Xiao & Zhu, 2013), and PSGM (Abubakar, Kumam, et al, 2020). In Matlab R2019b the four algorithms were programmed and run on a PC with a 2.40GHZ CPU processor and 8.00 GB RAM.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Spectral Gradient Projection Method for Sparse Signal Reconstruction in Compressive Sensing

Abubakar¹,

Muangchoo²,

Muhammad³

et al. 2020

MAS

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In this paper, a new spectral gradient direction is proposed to solve the l1 -regularized convex minimization problem. The spectral parameter of the proposed method is computed as a convex combination of two existing spectral parameters of some conjugate gradient method. Moreover, the spectral gradient method is applied to the resulting problem at each iteration without requiring the Jacobian matrix. Furthermore, the proposed method is shown to have converge globally under some assumptions. Numerically, the proposed method is efficient and robust in terms of its quality in reconstructing sparse signal and low computational cost compared to the existing methods.

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A note on the spectral gradient projection method for nonlinear monotone equations with applications

Cited by 47 publications

References 33 publications

A Family of Modified Spectral Projection Methods for Nonlinear Monotone Equations with Convex Constraint

A Family of Modified Spectral Projection Methods for Nonlinear Monotone Equations with Convex Constraint

A Family of Derivative-Free Conjugate Gradient Methods for Constrained Nonlinear Equations and Image Restoration

A Spectral Gradient Projection Method for Sparse Signal Reconstruction in Compressive Sensing

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