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Abstract: Conjugate gradient methods have been paid attention to, because they can be directly applied to large-scale unconstrained optimization problems. In order to incorporate second order information of the objective function into conjugate gradient methods, Dai and Liao (2001) proposed a conjugate gradient method based on the secant condition. However, their method does not necessarily generate a descent search direction. On the other hand, Hager and Zhang (2005) proposed another conjugate gradient method which always generates a descent search direction. In this paper, combining Dai–Liao’s idea and Hager–Zhang’s idea, we propose conjugate gradient methods based on secant conditions that generate descent search directions. In addition, we prove global convergence properties of the proposed methods. Finally, preliminary numerical results are given. [Copyright &y& Elsevier]