Abstract
This article presents a modified quadratic hybridization of the Polak–Ribiere–Polyak and Fletcher–Reeves conjugate gradient method for solving unconstrained optimization problems. Global convergence, with the strong Wolfe line search conditions, of the proposed quadratic hybrid conjugate gradient method is established. We also report some numerical results to show the competitiveness of the new hybrid method.
Highlights
IntroductionNonlinear conjugate gradient method is a very powerful technique for solving large scale unconstrained optimization problems min{f (x) : x ∈ Rn},
Nonlinear conjugate gradient method is a very powerful technique for solving large scale unconstrained optimization problems min{f (x) : x ∈ Rn}, (1)where f : Rn → R is a continuously differentiable function
More information on these line search methods and other line search methods can be found in the literature [9, 14, 25, 31, 34, 37, 39, 41]. We suggest another approach to get a new hybrid nonlinear conjugate gradient method
Summary
Nonlinear conjugate gradient method is a very powerful technique for solving large scale unconstrained optimization problems min{f (x) : x ∈ Rn},. Where yk−1 = gk − gk−1 and || · || denotes the Euclidean norm of vectors These were the first scalars βk for nonlinear conjugate gradient methods to be proposed. Researchers try to devise some new methods, which have the advantages of these two kinds of methods This has been done mostly by combining two or more βk parameters in the same conjugate gradient method to come up with hybrid methods. The step length αk is often chosen to satisfy certain line search conditions It is very important in the convergence analysis and implementation of conjugate gradient methods. The line search in the conjugate gradient methods is often based on the weak Wolfe conditions f (xk + αkdk) ≤ f (xk) + μαkgkT dk (4).
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