A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing

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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“…By continuity of ϕ and setting m → ∞ yields −ϕ T t0 d t0 ≤ 0, which contradicts (21). Hence, proved.…”

Section: Convergence Analysismentioning

confidence: 83%

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

Ibrahim

Kumam

2020

IEEE Access

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“…where the inequalities are obtained from (13), (31) and Assumption 3 respectively. By (32) and the assumption…”

Section: B a Proof Of Lemmamentioning

confidence: 99%

“…In recent years, many efforts have been put in finding the solution of non-smooth or semi-smooth system of The associate editor coordinating the review of this manuscript and approving it for publication was Yanbo Chen . (1), in which F is non-differentiable, such as [10]- [13]. By approximating the secant condition of Quasi-Newton methods, Barzilai and Borwein proposed the BB step size in [14], which has been widely used in the unconstrained problem.…”

Section: Introductionmentioning

confidence: 99%

A Modified Spectral Gradient Projection Method for Solving Non-linear Monotone Equations with Convex Constraints and Its Application

2020

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In this paper, we propose a derivative free algorithm for solving non-linear monotone equations with convex constraints. The proposed algorithm combines the method of spectral gradient and the projection method. We also modify the backtracking line search technique. The global convergence of the proposed method is guaranteed, under the mild conditions. Further, the numerical experiments show that the largescale non-linear equations with convex constraints can be effectively solved with our method. The L 1-norm regularized problems in signal reconstruction are studied by using our method.

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“…We note that the sequences {t k }, {u k }, {F(t k )}, and {F(u k )} are bounded from (25), (31), (27), and (28), respectively. In addition, from the Lipchitz continuity and (25), we have…”

Section: Global Convergencementioning

confidence: 99%

“…The method was also successfully used to solve the sparse signal in compressive sensing. Interested readers may refer to the following articles [21][22][23][24][25][26][27] for an overview of algorithms used for solving monotone operator equations.…”

Section: Introductionmentioning

confidence: 99%

Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing

Ibrahim

Abubakar

et al. 2020

Mathematics

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In this paper, we propose, analyze, and test an alternative method for solving the 1 -norm regularization problem for recovering sparse signals and blurred images in compressive sensing. The method is motivated by the recent proposed nonlinear conjugate gradient method of Tang, Li and Cui [Journal of Inequalities and Applications, 2020 (1), 27] designed based on the least-squares technique. The proposed method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and an 1 -norm regularization term. The search directions generated by the proposed method are descent directions. In addition, under the monotonicity and Lipschitz continuity assumption, we establish the global convergence of the method. Preliminary numerical results are reported to show the efficiency of the proposed method in practical computation.

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A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing

Cited by 59 publications

References 35 publications

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

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

A Modified Spectral Gradient Projection Method for Solving Non-linear Monotone Equations with Convex Constraints and Its Application

Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing

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