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An efficient hybrid conjugate gradient method for unconstrained optimization. (English) Zbl 1007.90065
Summary: Recently, we propose a nonlinear conjugate gradient method, which produces a descent search direction at every iteration and converges globally provided that the line search satisfies the weak Wolfe conditions. In this paper, we will study methods related to the new nonlinear conjugate gradient method. Specifically, if the size of the scalar ${\beta }_{k}$ with respect to the one in the new method belongs to some interval, then the corresponding methods are proved to be globally convergent; otherwise, we are able to construct a convex quadratic example showing that the methods need not converge. Numerical experiments are made for two combinations of the new method and the Hestenes-Stiefel conjugate gradient method. The initial results show that, one of the hybrid methods is especially efficient for the given test problems.

##### MSC:
 90C30 Nonlinear programming 49M37 Methods of nonlinear programming type in calculus of variations 65K05 Mathematical programming (numerical methods)