Appl Math Optim 43:87–101 (2001)
2001 Springer-Verlag New York Inc.
New Conjugacy Conditions and Related Nonlinear Conjugate
and L.-Z. Liao
State Key Laboratory of Scientiﬁc and Engineering Computing,
Institute of Computational Mathematics and Scientiﬁc/Engineering Computing,
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences,
Box 2719, Beijing 100080, The People’s Republic of China
Department of Mathematics, Hong Kong Baptist University,
Kowloon Tong, Kowloon, Hong Kong
Abstract. Conjugate gradient methods are a class of important methods for uncon-
strained optimization, especially when the dimension is large. This paper proposes a
newconjugacy condition, which considers an inexact line search scheme but reduces
to the old one if the line search is exact. Based on the new conjugacy condition, two
nonlinear conjugate gradient methods are constructed. Convergence analysis for the
two methods is provided. Our numerical results show that one of the methods is
very efﬁcient for the given test problems.
KeyWords. Unconstrained optimization, Conjugate gradient, Line search, Global
AMS Classiﬁcation. 65K, 90C.
Our problem is to minimize a function of n variables,
min f (x), x ∈ R
This research was supported in part by the Chinese NSF Grant 19801033 and Grant FRG/97-98/II-42
of Hong Kong Baptist University.