Abstract
The conjugate gradient method was an efficient technique for solving the unconstrained optimization problem. In this paper, we made a linear combination with parameters βk of the DY method and the HS method, and putted forward the hybrid method of DY and HS. We also proposed the hybrid of FR and PRP by the same mean. Additionally, to present the two hybrid methods, we promoted the Wolfe line search respectively to compute the step size αk of the two hybrid methods. With the new Wolfe line search, the two hybrid methods had descent property and global convergence property of the two hybrid methods that can also be proved.
Highlights
The conjugate gradient method The conjugate gradient method is a very important and efficient technique for solving large scale minimization problems, due to it can complete with lower storage and simple computation (Birgin and Martinez 2001)
We are concerned with the conjugate gradient methods for solving (1)
In this paper, we have proposed two hybrid conjugate gradient methods, respectively are the hybrid method of DY and HS, the hybrid method of FR and PRP
Summary
The conjugate gradient method The conjugate gradient method is a very important and efficient technique for solving large scale minimization problems, due to it can complete with lower storage and simple computation (Birgin and Martinez 2001).Consider an unconstrained minimization problem.min f (x), x ∈ Rn (1)where Rn denotes an n-dimensional Euclidean space and f : Rn −→ R1 is a continuously differentiable function. The FR method is the earliest nonlinear conjugate gradient method, under the exact line search, Powell (1977) pointed out the FR method could continuously produce small steps and have the global efficiency property.
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