The Projection Technique for Two Open Problems of Unconstrained Optimization Problems

Yuan, Gonglin; Wang, Xiaoliang; Zhou, Sheng

doi:10.1007/s10957-020-01710-0

Cited by 23 publications

(16 citation statements)

References 71 publications

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“…There are two main reasons why these methods are efficient for solving unconstrained optimization problems: low memory requirement and strong local and global convergence properties. In addition, they do not require any matrix storage and are suitable to solve large-scale optimization problems, see [8,17,21,22].…”

Section: Related Workmentioning

confidence: 99%

A new family of Polyak and Ribiere conjugate gradient method for impulse noise removal

Mousavi

Esmaeilpour

Sheikhahmadi

2022

Preprint

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In this paper, impulse noise removal problem is formulated as an unconstrained optimization problem with smooth objective function. It can be solved by conjugate gradient methods with desired properties (low memory and strong global convergence) in high dimensions. Accordingly, a family of the Polyak-Ribi`ere conjugate gradient directions is constructed for which the descent condition holds. Using several images, it is shown that the new methods are very robust and efficient in comparison with other competitive methods for impulse noise removal, especially in terms of the peak signal to noise ratio (PSNR). Mathematics Subject Classification (2000) 90C30a · 90C25 · 90C90 · 68U10 · 03D15

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Section: Related Workmentioning

confidence: 99%

A new family of Polyak and Ribiere conjugate gradient method for impulse noise removal

Mousavi

Esmaeilpour

Sheikhahmadi

2022

Preprint

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“…e idea of the projection can also be found in [6,8,35]. Based on the above discussions, the modified algorithm is given in Algorithm 1.…”

Section: Motivation and Algorithmmentioning

confidence: 99%

“…where x k is the current point, s k � x k+1 − x k � α k d k , α k is a step size, and d k is a search direction at x k . ere exist many algorithms for (1) [1][2][3][4][5][6][7][8][9]. Davidon [10] pointed out that the quasi-Newton method is one of the most effective methods for solving nonlinear optimization problems.…”

Section: Introductionmentioning

confidence: 99%

The Global Convergence of a Modified BFGS Method under Inexact Line Search for Nonconvex Functions

Feng

2021

Mathematical Problems in Engineering

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Among the quasi-Newton algorithms, the BFGS method is often discussed by related scholars. However, in the case of inexact Wolfe line searches or even exact line search, the global convergence of the BFGS method for nonconvex functions is not still proven. Based on the aforementioned issues, we propose a new quasi-Newton algorithm to obtain a better convergence property; it is designed according to the following essentials: (1) a modified BFGS formula is designed to guarantee that B k + 1 inherits the positive definiteness of B k ; (2) a modified weak Wolfe–Powell line search is recommended; (3) a parabola, which is considered as the projection plane to avoid using the invalid direction, is proposed, and the next point x k + 1 is designed by a projection technique; (4) to obtain the global convergence of the proposed algorithm more easily, the projection point is used at all the next iteration points instead of the current modified BFGS update formula; and (5) the global convergence of the given algorithm is established under suitable conditions. Numerical results show that the proposed algorithm is efficient.

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“…The engineering Muskingum model and image restoration problems were used to determine the interesting aspects of the given algorithm [30]. The generalized conjugate gradient algorithms were studied for solving large-scale unconstrained optimization problems within the real world applications, and two open problems were formulated [31][32][33].…”

Section: Introductionmentioning

confidence: 99%

A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems

et al. 2021

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A Polak–Ribière–Polyak (PRP) algorithm is one of the oldest and popular conjugate gradient algorithms for solving nonlinear unconstrained optimization problems. In this paper, we present a q-variant of the PRP (q-PRP) method for which both the sufficient and conjugacy conditions are satisfied at every iteration. The proposed method is convergent globally with standard Wolfe conditions and strong Wolfe conditions. The numerical results show that the proposed method is promising for a set of given test problems with different starting points. Moreover, the method reduces to the classical PRP method as the parameter q approaches 1.

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The Projection Technique for Two Open Problems of Unconstrained Optimization Problems

Cited by 23 publications

References 71 publications

A new family of Polyak and Ribiere conjugate gradient method for impulse noise removal

A new family of Polyak and Ribiere conjugate gradient method for impulse noise removal

The Global Convergence of a Modified BFGS Method under Inexact Line Search for Nonconvex Functions

A q-Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization problems

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