The conjugate gradient method is one of the most useful and the earliest-discovered techniques for solving large-scale nonlinear optimization problems. Many variants of this method have been proposed, and some are widely used in practice. In this article, we study the descent Dai–Yuan conjugate gradient method which guarantees the sufficient descent condition for any line search. With exact line search, the introduced conjugate gradient method reduces to the Dai–Yuan conjugate gradient method. Finally, a global convergence result is established when the line search fulfils the Goldstein conditions.
Al-BaaliM (1985) Descent property and global convergence of the Fletcher–Reeves method with inexact line search. IMA Journal of Numerical Analysis5: 121–124.
2.
AndreiN (2008) A Dai–Yuan conjugate gradient algorithm with sufficient descent and conjugacy conditions for unconstrained optimization. Applied Mathematics Letters21: 165–171.
3.
ArmijoL (1966) Minimization of functions having Lipschits continuous partial derivatives. Pacific Journal of Mathematics16: 1–3.
4.
DaiYH (2003) A family of hybrid conjugate gradient methods for unconstrained optimization. Mathematics of Computation72: 1317–1328.
5.
DaiYHYuanY (1999) A nonlinear conjugate gradient method with a strong global convergence property. SIAM Journal on Optimization10: 177–182.
6.
DaiYHYuanY (1996) Convergence properties of the conjugate descent method. Advances in Mathematics25: 552–562.
7.
DaiYHYuanY (2000) A nonlinear conjugate gradient with a strong global convergence properties. SIAM Journal on Optimization10: 177–182.
8.
DaiYHYuanY (2003) A class of globally convergent conjugate gradient methods. Science in China Series A: Mathematics46: 251–262.
9.
Fletcher R (1997) Practical Method of Optimization, 2nd edn (Unconstrained Optimization, Vol. I). New York: Wiley.
10.
FletcherRReevesC (1964) Function minimization by conjugate gradients. Computer Journal7: 149–154.
11.
HagerWWZhangH (2005) A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM Journal on Optimization16: 170–192.
12.
HestenesMRStiefelE (1952) Method of conjugate gradient for solving linear equations. Journal of Research National Bureau of Standards49: 409–436.
13.
LiuYStoreyC (1992) Efficient generalized conjugate gradient algorithms, part 1: theory. Journal of Optimization Theory and Applications69: 129–137.
14.
NocedalJ (1992) Theory of algorithms for unconstrained optimazation. Acta Numerica1: 199–242.
15.
NocedalJWrightJS (1999) Numerical Optimization. New York: Springer-Verlag.
16.
PolakE (1997) Optimazation: Algorithms and Consistent Approximations. New York: Springer-Verlag.
17.
PolyakBT (1969) The conjugate gradient method in extreme problems. USSR Computational Mathematics and Mathematical Physics9: 94–112.
18.
Powell MJD (1984) Nonconvex Minimization Calculations and the Conjugate Gradient Method (Lecture Notes in Mathematics, Vol. 1066). Berlin: Springer-Verlag, pp. 122–141.
19.
ShiZJShenJ (2007) Convergence of Liu–Storey conjugate gradient method. European Journal of Operational Research182: 552–560.
20.
YabeH (2004) Global convergence properties of nonlinear conjugate gradient methods with modified secant condition. Computational Optimization and Applications28: 203–225.
21.
YuGGuanLChenW (2008) Spectral conjugate gradient methods with sufficient descent property for largescale unconstrained optimization. Optimization Methods and Software23: 275–293.
22.
YuanY (1993) Numerical Methods for Nonlinear Programming. Shanghai: Shanghai Scientific and Technical Publishers.
