A conjugate gradient projection method for solving equations with convex constraints

Zheng, Lianxi; Yang, Lei; Liang, Yong

doi:10.1016/j.cam.2020.112781

“…The efficiency and suitability of the proposed algorithm for large scale problems is as a result of its low storage and lack of Jacobian matrix computation. Numerical results obtained from the problems considered proved that this algorithm is more efficient in comparison with some existing algorithms, specifically, those in [45], [49] and [50]. Moreover, the global convergence properties of the algorithm are proved under some appropriate assumptions.…”

Section: Discussionmentioning

confidence: 87%

“…Using some test problems, we compare the performance of MCDPM with three other algorithms. Specifically, Algorithm 2.1 proposed by Li et al [49], Self-adaptive three-term conjugate gradient method (SATCGM) algorithm by Wang et al [50] and the NLS algorithm proposed in [45]. In MCDPM, Algorithm 2.1 and SATCGM, Solodov and Svaiter line search [34] is used whereas the NLS algorithm considered a different line search.…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

A Modified Conjugate Descent Projection Method for Monotone Nonlinear Equations and Image Restoration

Aji

¹

,

Siricharoen

²

,

Abubakar

³

et al. 2020

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In this paper, we propose a modified conjugate descent (CD) projection algorithm for solving system of nonlinear monotone equations with convex constraints. The search direction in this algorithm use a convex combination of the steepest descent algorithm and the well-known CD method. The algorithm proves to be quite efficient for solving large scale monotone nonlinear equations, as it has low storage requirement and does not need the computation of Jacobian matrix. We prove the convergence of the algorithm using some conditions, and perform numerical experiments on some test problems. In order to show the efficiency of our proposed algorithm, the numerical performance is compared with some existing algorithms. Finally, by reformulating 1 regularized problem into monotone equation, we successfully apply the algorithm to restore some blurred images. The numerical results obtained prove that the algorithm can be used as an efficient and qualitative solver for image restoration problems.

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“…In NLS, Algorithm 2.1 and SATCGM all parameters are maintained as they are in [45], [49], [50] respectively. As for MCDPM, σ = 0.0001 is used whereas for β, different values were tested in the interval (0, 1), but β = 0.6 was observed to give the best result.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Two Hybrid Spectral Methods With Inertial Effect for Solving System of Nonlinear Monotone Equations With Application in Robotics

Aji

¹

,

Awwal

²

,

Yahaya

³

et al. 2021

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In this paper, we propose a modified conjugate descent (CD) projection algorithm for solving system of nonlinear monotone equations with convex constraints. The search direction in this algorithm use a convex combination of the steepest descent algorithm and the well-known CD method. The algorithm proves to be quite efficient for solving large scale monotone nonlinear equations, as it has low storage requirement and does not need the computation of Jacobian matrix. We prove the convergence of the algorithm using some conditions, and perform numerical experiments on some test problems. In order to show the efficiency of our proposed algorithm, the numerical performance is compared with some existing algorithms. Finally, by reformulating 1 regularized problem into monotone equation, we successfully apply the algorithm to restore some blurred images. The numerical results obtained prove that the algorithm can be used as an efficient and qualitative solver for image restoration problems.

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“…In this section, the numerical behavior of the proposed algorithm (Algorithm 1) in comparison with two existing methods is examined. We compare the performance of Algorithm 1 with a conjugate gradient projection method for solving nonlinear equations with convex constraints by Zheng et al [46] denoted as Algorithm 2 and a new three-term conjugate gradient-based projection method for solving large-scale nonlinear monotone equations by Koorapetse et al [24] denoted as Algorithm 3. Algorithm 1 is implemented using the following parameters: σ = 0.001, µ = 1 ρ = 0.7, γ = 1.7 and λ = 1.2.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A New Three-Term Hestenes-Stiefel Type Method for Nonlinear Monotone Operator Equations and Image Restoration

Abubakar

¹

,

Muangchoo

²

,

Ibrahim

³

et al. 2021

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In this article, a derivative-free method of Hestenes-Stiefel type is proposed for solving system of monotone operator equations with convex constraints. The method proposed is matrix-free, and its sequence of search directions are bounded and satisfies the sufficient descent condition. The global convergence of the proposed approach is established under the assumptions that the underlying operator is monotone and Lipschitz continuous. Numerical experiment results are reported to show the efficiency of the proposed method. Furthermore, to illustrate the applicability of the proposed method, it is used in restoring blurred images.INDEX TERMS Derivative-free algorithm, Monotone operator equations, projection technique, image restoration.

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A conjugate gradient projection method for solving equations with convex constraints

Cited by 27 publications

References 32 publications

A Modified Conjugate Descent Projection Method for Monotone Nonlinear Equations and Image Restoration

A Modified Conjugate Descent Projection Method for Monotone Nonlinear Equations and Image Restoration

Two Hybrid Spectral Methods With Inertial Effect for Solving System of Nonlinear Monotone Equations With Application in Robotics

A New Three-Term Hestenes-Stiefel Type Method for Nonlinear Monotone Operator Equations and Image Restoration

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