Abstract
In this paper, we propose a Dai–Liao (DL) conjugate gradient method for solving large-scale system of nonlinear equations. The method incorporates an extended secant equation developed from modified secant equations proposed by Zhang et al. (J Optim Theory Appl 102(1):147–157, 1999) and Wei et al. (Appl Math Comput 175(2):1156–1188, 2006) in the DL approach. It is shown that the proposed scheme satisfies the sufficient descent condition. The global convergence of the method is established under mild conditions, and computational experiments on some benchmark test problems show that the method is efficient and robust.
Highlights
A typical system of nonlinear equations has the general form F (x) = 0, (1)where F : Rn → Rn is a nonlinear mapping assumed to be continuously differentiable in a neighborhood of Rn
We describe the algorithm of the proposed method as follows: Algorithm 2.4 A Dai–Liao conjugate gradient (CG) method (ADLCG)
We prove the global convergence of the ADLCG method
Summary
Where F : Rn → Rn is a nonlinear mapping assumed to be continuously differentiable in a neighborhood of Rn. CG methods based on modified secant equations have been studied by Narushima and Yabe [57] and Reza Arazm et al [7] These methods have been found to be numerically efficient and globally convergent under suitable conditions, but like the DL method, they fail to ensure sufficient descent. As a further research of the Perry’s conjugate gradient method, Dai et al [21] combined the modified Perry conjugate gradient method [41] and the hyperplane projection technique of Solodov and Svaiter [48] to propose a derivative-free method for solving large-scale nonlinear monotone equations. Based on the work of Babaie-Kafaki and Ghanbari [9], and the Dai–Liao (DL) [18] approach, we propose a Dai–Liao conjugate gradient method for system of nonlinear equations by incorporating an extended secant equation in the classical DL update.
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