## 1 Introduction  
Conjugate gradient methods are among the most popular methods for solving optimization problem, especially for large-scale problems due to the simplicity and low storage of their iterative form [1]. Consider the following unconstrained optimization problem:  
\[\min\{f(x):x\in R^{n}\}, \tag{1}\]  
where \(f:R^{n}\to R\) is continuously differentiable. Let \(x_{0}\) be any initial point of the solution of the problem (1), then the conjugate gradient method generates an iteration sequence as follows:  
\[x_{k+1}=x_{k}+a_{k}d_{k},k=0,1,2,... \tag{2}\]  
where \(x_{k}\) is the \(k\)th iterative point, \(a_{k}>0\) is a steplength which is obtained by carrying out some line search, and \(d_{k}\) is a search direction defined by  
\[d_{k}=\left\{ \begin{array}{ll}-g_{k},&\mbox{if }\ k=0,\\ -g_{k}+\beta_{k}d_{k-1},&\mbox{if }\ k\geq 1,\end{array} \right. \tag{3}\]  
where \(g_{k}=g(x_{k})\) denotes the gradient of the function \(f(x)\) at \(x_{k}\) and \(\beta_{k}\) is a scalar that determines different conjugate gradient methods [2; 3; 4; 5; 6]. In this paper, we focus our attention on well-known methods such as Polak-Ribiere-Polyak (PRP) [4; 5], Hestenes-Stiefel (HS) [3] and Liu-Storey (LS) [6] methods which share the same numerator \(g_{k}^{T}y_{k-1}\) in \(\beta_{k}\). The update parameters of these methods are, respectively, given by  
\[\beta_{k}^{PRP}=\frac{g_{k}^{T}y_{k-1}}{\|g_{k-1}\|^{2}}\,,\beta_{k}^{HS}= \frac{g_{k}^{T}y_{k-1}}{d_{k-1}^{T}y_{k-1}}\,,\beta_{k}^{LS}=-\frac{g_{k}^{T}y _{k-1}}{d_{k-1}^{T}\beta_{k-1}}, \tag{4}\]where \(y_{k-1}=g_{k}-g_{k-1},\|\cdot\|\) denotes the Euclidean norm of vectors. Other nonlinear conjugate gradient methods and their global convergence can be found in [1; 7]. It is well-known that the PRP, HS and LS methods are generally regarded to be the most efficient methods in practical computation. This can be attributed to the property (*), which was derived in [PERSON] and [PERSON] [8]. [PERSON] [4] obtained the global convergence of the PRP method for the strongly convex functions with exact line search. [PERSON] [9] also obtained the global convergence of the PRP method under the assumption that the search direction satisfies a descent condition:  
\[g_{k}^{T}d_{k}<0,\forall k\geq 0. \tag{5}\]  
and the following standard Wolfe line search  
\[\left\{ \begin{array}{l}f(x_{k}+\alpha_{k}d_{k})-f(x_{k})\leq\delta\alpha_{k} g_{k}^{T}d_{k},\\ g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\geq\sigma_{k}^{T}d_{k},\end{array} \right. \tag{6}\]  
where \(0<\delta\leq\sigma<1\). Their convergence properties are not so good in many situations. [PERSON] [10] gave a counter example which showed that there exist nonconvex functions on which the PRP method does not converge globally even if the exact line search is used. Inspired by [PERSON]'s work, [PERSON] and [PERSON] [8] proved that the modified PRP method is globally convergent in which \(\beta_{k}\) is given by \(\beta_{k}^{PRP+}=max\{\beta_{k}^{PRP},0\}\). The search direction prevents effectively jamming phenomena from occurring and satisfies the descent property (5) or the following sufficient descent condition:  
\[g_{k}^{T}d_{k}\leq-c\|g_{k}\|^{2},\forall k\geq 0,c>0, \tag{7}\]  
which is very important for establishing the global convergence of the proposed method. In [11], [PERSON] and [PERSON] proposed a modified HS formula for \(\beta_{k}\) defined by  
\[\beta_{k}^{HZ}=\beta_{k}^{HS}-2\frac{\|y_{k}\|^{2}g_{k}^{T}d_{k-1}}{(d_{k-1}^{ T}y_{k-1})^{2}}.\]  
More specifically, in their proposed method, called CG-DESCENT. They showed that the method possesses the sufficient descent property with \(\mathrm{c}=7/8\). Afterwards, they also presented the following extension of \(\beta_{k}^{HZ}\):  
\[\beta_{k}^{HZ}=\beta_{k}^{HS}-\theta_{k}\frac{\|y_{k}\|^{2}g_{k}^{T}d_{k-1}}{( d_{k-1}^{T}y_{k-1})^{2}},\]  
where \(\theta_{k}\) is a nonnegative parameter. If \(\theta_{k}\geq\theta>1/4\), then the method possesses the sufficient descent property with \(c=1-1/(4\theta)\).