scholarly journals Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
San-Yang Liu ◽  
Yuan-Yuan Huang ◽  
Hong-Wei Jiao
Keyword(s):  
Global Convergence ◽  
Projection Method ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Monotone Equations ◽  
Mild Conditions ◽  
Sufficient Descent ◽  
Nonlinear Monotone Equations

Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.

scholarly journals A modified Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations

2021 ◽  
Author(s):  
Mompati Koorapetse ◽  
P Kaelo ◽  
S Kooepile-Reikeletseng
Keyword(s):  
Global Convergence ◽  
Projection Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Numerical Results ◽  
Projection Technique ◽  
Monotone Equations ◽  
Derivative Free ◽  
Nonlinear Monotone Equations

In this paper, a new modified Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations is presented. The method is developed by combining a modified Perry's conjugate gradient method with the hyperplane projection technique. Global convergence and numerical results of the proposed method are established. Preliminary numerical results show that the proposed method is promising and efficient compared to some existing methods in the literature.

GLOBAL CONVERGENCE OF TWO KINDS OF THREE-TERM CONJUGATE GRADIENT METHODS WITHOUT LINE SEARCH

2013 ◽  
Vol 30 (01) ◽  
pp. 1250043
Author(s):  
LIANG YIN ◽  
XIONGDA CHEN
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Search Procedure ◽  
Computational Results ◽  
Large Scale Problems ◽  
Sufficient Descent

The conjugate gradient method is widely used in unconstrained optimization, especially for large-scale problems. Recently, Zhang et al. proposed a three-term PRP method (TTPRP) and a three-term HS method (TTHS), both of which can produce sufficient descent conditions. In this paper, the global convergence of the TTPRP and TTHS methods is studied, in which the line search procedure is replaced by a fixed formula of stepsize. This character is of significance when the line search is expensive in some particular applications. In addition, relevant computational results are also presented.

scholarly journals A NEW THREE-TERM CONJUGATE GRADIENT-BASED PROJECTION METHOD FOR SOLVING LARGE-SCALE NONLINEAR MONOTONE EQUATIONS

2019 ◽  
Vol 24 (4) ◽  
pp. 550-563
Author(s):  
Mompati Koorapetse ◽  
Professor Kaelo
Keyword(s):  
Projection Method ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Sufficient Descent Condition ◽  
Monotone Equations ◽  
Derivative Free ◽  
Gradient Based ◽  
Sufficient Descent ◽  
Descent Condition ◽  
Nonlinear Monotone Equations

A new three-term conjugate gradient-based projection method is presented in this paper for solving large-scale nonlinear monotone equations. This method is derivative-free and it is suitable for solving large-scale nonlinear monotone equations due to its lower storage requirements. The method satisfies the sufficient descent condition FTkdk ≤ −τ‖Fk‖2, where τ > 0 is a constant, and its global convergence is also established. Numerical results show that the method is efficient and promising.

scholarly journals A New Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Bakhtawar Baluch ◽  
Zabidin Salleh ◽  
Ahmad Alhawarat
Keyword(s):  
Global Convergence ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Sufficient Descent Property ◽  
Descent Property ◽  
Sufficient Descent

This paper describes a modified three-term Hestenes–Stiefel (HS) method. The original HS method is the earliest conjugate gradient method. Although the HS method achieves global convergence using an exact line search, this is not guaranteed in the case of an inexact line search. In addition, the HS method does not usually satisfy the descent property. Our modified three-term conjugate gradient method possesses a sufficient descent property regardless of the type of line search and guarantees global convergence using the inexact Wolfe–Powell line search. The numerical efficiency of the modified three-term HS method is checked using 75 standard test functions. It is known that three-term conjugate gradient methods are numerically more efficient than two-term conjugate gradient methods. Importantly, this paper quantifies how much better the three-term performance is compared with two-term methods. Thus, in the numerical results, we compare our new modification with an efficient two-term conjugate gradient method. We also compare our modification with a state-of-the-art three-term HS method. Finally, we conclude that our proposed modification is globally convergent and numerically efficient.

scholarly journals Two New Conjugate Gradient Methods for Unconstrained Optimization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Meixing Liu ◽  
Guodong Ma ◽  
Jianghua Yin
Keyword(s):  
Unconstrained Optimization ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Gradient Methods ◽  
Conjugate Gradient Methods ◽  
Globally Convergent ◽  
Sufficient Descent ◽  
Strong Wolfe Line Search ◽  
Wolfe Line Search

The conjugate gradient method is very effective in solving large-scale unconstrained optimal problems. In this paper, on the basis of the conjugate parameter of the conjugate descent (CD) method and the second inequality in the strong Wolfe line search, two new conjugate parameters are devised. Using the strong Wolfe line search to obtain the step lengths, two modified conjugate gradient methods are proposed for general unconstrained optimization. Under the standard assumptions, the two presented methods are proved to be sufficient descent and globally convergent. Finally, preliminary numerical results are reported to show that the proposed methods are promising.

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