Abstract
It is well known that global convergence has not been established for the Hestenes-Stiefel (HS) conjugate gradient method using the traditional line searches conditions. In this paper, under some suitable conditions, by using a modified Armijo line search, global convergence results were established for the HS method. Preliminary numerical results on a set of large-scale problems were reported to show that the HS method’s computational effiiciency is encouraging.
Highlights
Consider the unconstrained optimization problem{min f (x), x ∈ Rn}, 1 (1)where f : Rn −→ R is continuously differentiable
In this paper we propose a new Armijo line search in which an appropriate initial step size s is defined and varies at each iteration
It is possible that the initial choice of step size (38) is reasonable for the SCG method in practical computation.All the facts show that choosing an adequate initial step size at each iteration is very important for line search methods, especially for conjugate gradient methods
Summary
Where f : Rn −→ R is continuously differentiable. The line search method usually takes the following iterative formula xk+1 = xk + αkdk. Al-Baali [1], Toouati-Ahmed and Storey [3], Hu and Storey [13], Gilbert and Nocedal [10] analyzed the global convergence of algorithms related to the Fletcher– Reeves method with the strong Wolfe line search Their convergence analysis used the sufficient descent condition (1.8). The new Armijo line search enables us to find the step size αk at each iteration and guarantees the global convergence of the original HS conjugate gradient method under some mild conditions. The global convergence and linear convergence rate are analyzed and numerical results show that HS method with the new Armijo line search is more effective than other similar methods in solving large scale minimization problems
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