A projection method for a system of nonlinear monotone equations with convex constraints

Wang, Chuanwei; Wang, Yiju; Xu, Chuanliang

doi:10.1007/s00186-006-0140-y

Cited by 104 publications

(65 citation statements)

References 13 publications

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“…To remove the above mentioned restricted conditions and motivated by the projection method for nonlinear equations [23,24,27,28], especially the technique in [27], in this paper, we design a perturbed spectral projected gradient method for problem (1.1). Here the perturbation means that the gradient of the objective function is computed or supplied with some error.…”

Section: Then the Secant Equation Becomesmentioning

confidence: 99%

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Yu¹,

Gan²

2016

J. Nonlinear Sci. Appl.

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In this paper, we propose a spectral projected gradient method for the convex optimization problem with singular solution. By solving the equivalent equation of the gradient function, this method combines the perturbed spectral gradient direction with the projection direction to generate the next iteration point. Under some mild conditions, we establish the global convergence and the local R-linear convergence rate under the local error bound condition. Preliminary numerical tests are given to show that the proposed method works well.

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Section: Then the Secant Equation Becomesmentioning

confidence: 99%

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Yu¹,

Gan²

2016

J. Nonlinear Sci. Appl.

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show abstract

Section: Introductionmentioning

confidence: 99%

“…Recently, the literature [15] proposed a projection method for solving problem (1), which possesses a very nice global convergence property without the differentiability or locally Lipschitz continuity assumptions. The numerical performances given in [15] show that the projection method for solving problem (1) is really efficient and has strong stability. More recently, the literatures [9,14] proposed, respectively, a modified version for the method by changing the projection region in order to accelerate the convergence rate.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, the literatures [9,14] proposed, respectively, a modified version for the method by changing the projection region in order to accelerate the convergence rate. However, the literatures [9,15,14] require the mapping F is monotone, which seems too stringent a requirement for the purpose of ensuring global convergence property of the projection method.…”

Section: Introductionmentioning

confidence: 99%

“…In the proof of global convergence of our method, the underlying mapping F need only to satisfy the property (2) which is much weaker than monotone or more generally pseudomontone. Moreover, under weaker assumptions than those of [12,15,14], the local rate of convergence of the iterative sequence is established.…”

Section: Introductionmentioning

confidence: 99%

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A New Projection Algorithm for Solving a System of Nonlinear Equations With Convex Constraints

Lian¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. We present a new algorithm for solving a system of nonlinear equations with convex constraints which combines proximal point and projection methodologies. Compared with the existing projection methods for solving the problem, we use a different system of linear equations to obtain the proximal point; and moreover, at the step of getting next iterate, our projection way and projection region are also different. Based on the Armijo-type line search procedure, a new hyperplane is introduced. Using the separate property of hyperplane, the new algorithm is proved to be globally convergent under much weaker assumptions than monotone or more generally pseudomonotone. We study the convergence rate of the iterative sequence under very mild error bound conditions.

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An improved Dai‐Liao‐style hybrid conjugate gradient‐based method for solving unconstrained nonconvex optimization and extension to constrained nonlinear monotone equations

Yuan,

Shao,

Zeng

et al. 2024

Math Methods in App Sciences

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In this work, for unconstrained optimization, we introduce an improved Dai‐Liao‐style hybrid conjugate gradient method based on the hybridization‐based self‐adaptive technique, and the search direction generated fulfills the sufficient descent and trust region properties regardless of any line search. The global convergence is established under standard Wolfe line search and common assumptions. Then, combining the hyperplane projection technique and a new self‐adaptive line search, we extend the proposed conjugate gradient method and obtain an improved Dai‐Liao‐style hybrid conjugate gradient projection method to solve constrained nonlinear monotone equations. Under mild conditions, we obtain its global convergence without Lipschitz continuity. In addition, the convergence rates for the two proposed methods are analyzed, respectively. Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

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A projection method for a system of nonlinear monotone equations with convex constraints

Cited by 104 publications

References 13 publications

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

A New Projection Algorithm for Solving a System of Nonlinear Equations With Convex Constraints

An improved Dai‐Liao‐style hybrid conjugate gradient‐based method for solving unconstrained nonconvex optimization and extension to constrained nonlinear monotone equations

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