Conjugate gradient (CG) algorithms have played special roles in solving large scale nonlinear optimization problems with smooth objective functions f : Rn ! R. Search directions in the CG algorithms are generated by the sequence d1 = ¡rf(x1) and dk = ¡rf(xk) + ¯kdk¡1, for k ¸ 2. By introducing di®erent conjugacy conditions, researchers proposed di®erent formulas for ¯k. The related CG algorithms may have quite di®erent behaviors for general functions. Recently, Dai and Liao [1] proposed some new formulas for ¯k based on the standard secant equation. On the basis of the idea proposed by Dai and Liao, researchers made some e®orts to obtain new formulas for ¯k [2, 4, 5]. Here, we ¯rst make a modi¯cation on the secant equation proposed by Zhang and Xu [3], and then, using our modi¯ed secant equation and Dai-Liao's approach, we propose a new conjugacy condition and obtain a new formula for ¯k. It can be shown that under some proper conditions our CG algorithm is globally convergent for general functions. Numerical results showed that our algorithm is competitive and sometimes preferable to some recently proposed CG algorithms.

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