Abstract
In this paper, three new hybrid nonlinear conjugate gradient methods are presented, which produce suf?cient descent search direction at every iteration. This property is independent of any line search or the convexity of the objective function used. Under suitable conditions, we prove that the proposed methods converge globally for general nonconvex functions. The numerical results show that all these three new hybrid methods are efficient for the given test problems.
Highlights
IntroductionWe consider the unconstrained optimization problem: min f x , x Rn (1.1)
In this paper, we consider the unconstrained optimization problem: min f x, x Rn (1.1)where f : Rn R is continuously differentiable and the gradient of f at x is denoted by g x .Due to its simplicity and its very low memory requirement, the conjugate gradient (CG) method plays a very important role for solving (1.1)
Powell [12] gave a counter example which showed that there exist nonconvex functions such that the PRP method may cycle and does not approach any stationary point even with exact line search. one would be satisfied with its global convergence, the FR method performs much worse than the PRP (HS, LS) method in real computations
Summary
We consider the unconstrained optimization problem: min f x , x Rn (1.1). In the convergence analysis of conjugate gradient methods, one hopes the inexact line search such as the. Powell [12] gave a counter example which showed that there exist nonconvex functions such that the PRP method may cycle and does not approach any stationary point even with exact line search. one would be satisfied with its global convergence, the FR method performs much worse than the PRP (HS, LS) method in real computations. Combining the good numerical performance of the PRP and HS methods and the nice global convergence properties of the FR and DY methods, recently, [13] and [14] proposed some hybrid methods which we call the H1 method and the H2 method, respectively, that is, H1 k max.
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