Abstract
A hybrid method combining the FR conjugate gradient method and the WYL conjugate gradient method is proposed for unconstrained optimization problems. The presented method possesses the sufficient descent property under the strong Wolfe-Powell (SWP) line search rule relaxing the parameterσ<1. Under the suitable conditions, the global convergence with the SWP line search rule and the weak Wolfe-Powell (WWP) line search rule is established for nonconvex function. Numerical results show that this method is better than the FR method and the WYL method.
Highlights
Consider the following n variables unconstrained optimization problem: min f x, x∈Rn1.1 where f : Rn → R is smooth and its gradient g x is avaible
The sufficient descent condition gkT dk ≤ −c gk 2, ∀k ≥ 0, 1.6 where c > 0 is a constant, is crucial to insure the global convergence of the nonlinear conjugate gradient method 23, 26–28
It is well known that there exist many new conjugate gradient methods see 1, 13–16, 18, 19, 29, 30 which have good properties and good numerical performances
Summary
College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China. A hybrid method combining the FR conjugate gradient method and the WYL conjugate gradient method is proposed for unconstrained optimization problems. The presented method possesses the sufficient descent property under the strong Wolfe-Powell SWP line search rule relaxing the parameter σ < 1. The global convergence with the SWP line search rule and the weak Wolfe-Powell WWP line search rule is established for nonconvex function. Numerical results show that this method is better than the FR method and the WYL method.
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