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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 394096, 9 pages
Research Article

Two Modified Three-Term Type Conjugate Gradient Methods and Their Global Convergence for Unconstrained Optimization

1College of Communication Engineering, Jilin University, Changchun 130025, China
2Department of Mathematics, College of Humanities and Sciences of Northeast Normal University, Changchun 130117, China
3Key Laboratory of Bionic Engineering of Ministry of Education, Jilin University, Changchun 130025, China

Received 22 July 2014; Revised 5 November 2014; Accepted 6 November 2014; Published 23 November 2014

Academic Editor: Masoud Hajarian

Copyright © 2014 Zhongbo Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Two modified three-term type conjugate gradient algorithms which satisfy both the descent condition and the Dai-Liao type conjugacy condition are presented for unconstrained optimization. The first algorithm is a modification of the Hager and Zhang type algorithm in such a way that the search direction is descent and satisfies Dai-Liao’s type conjugacy condition. The second simple three-term type conjugate gradient method can generate sufficient decent directions at every iteration; moreover, this property is independent of the steplength line search. Also, the algorithms could be considered as a modification of the MBFGS method, but with different . Under some mild conditions, the given methods are global convergence, which is independent of the Wolfe line search for general functions. The numerical experiments show that the proposed methods are very robust and efficient.