###### Abstract
The author used Equation (7) to establish the convergence properties of \(\beta_{k}^{FR}\), which are defined as follows: There exists a constant \(c>0\) such that  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]0  
In this case, the function is quadratic, i.e., \(f(x)=g^{T}x+(1/2)x^{T}Hx\) and the step size obtained by exact line search (3) indicates that the CG method satisfies the conjugacy condition, i.e., \(d_{i}^{T}Hd_{j}^{T}=0\), \(\forall i\
eq j\). Using the mean value theorem and exact line search of Equation (2), we can obtain \(\beta_{k}^{HS}\). From the quasi-Newton method, BFGS method, the limited memory (LBFGS) method, and using Equation (2), [PERSON] and [PERSON] [12] proposed the following conjugacy condition. \[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]1  
where \(s_{k-1}=x_{k}-x_{k-1}\), and \(t\geq 0\), [12] use \(t=0.1\) in the numerical experiments. In the case of \(t=0\), Equation (8) becomes the classical conjugacy condition. By using Equations (2) and (8), [PERSON] and [PERSON] [12] proposed the following CG formula  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]2  
However, \(\beta_{k}^{DL}\) face the same problem as \(\beta_{k}^{PRP}\) and \(\beta_{k}^{HS}\), i.e., \(\beta_{k}^{DL}\) is not non-negative in general. Thus, [PERSON] and [PERSON] [12] replaced Equation (9) by  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]3  
[PERSON] and [PERSON] [13; 14] presented a modified CG parameter that satisfies the descent property for any inexact line search with \(g_{k}^{T}d_{k}\leq-(7/8)\|g_{k}\|^{2}\). This new version of the CG method is convergent whenever the line search satisfies the WP line search. This formula is given as follows:  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]4  
where \(\beta_{k}^{N}=\frac{1}{d_{k}^{T}y_{k}}(y_{k}-2d_{k}\frac{\|y_{k}\|^{2}}{d_{k} ^{T}y_{k}})\)\(g_{k}\), \(\eta_{k}=-\frac{1}{\|g_{k}\|\min\{\eta_{r},\ \|g_{k}\|\}}\), and \(\eta>0\) is a constant. Note that if \(t=2\frac{\|y_{k}\|^{2}}{s_{k}^{2}y_{k}}\), then \(\beta_{k}^{N}=\beta_{k}^{DY}\). [PERSON] et al. [15] and [PERSON] and [PERSON] [16] used [PERSON]'s series and presented the following modified secant relation:  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]5 \[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]6  
From (12), [PERSON] and [PERSON] [17] proposed the following formula:  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]7  
Moreover, [PERSON] et al. [18] proposed the following formula:  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]8  
[PERSON] et al. [19] proposed the CG method by replacing \(d_{k-1}\) by \(g_{k-1}\) in \(\beta_{k}^{HS}\) as follows:  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]9  
Furthermore, [PERSON] et al. [20] proposed a non-negative formula, referred to as the WYL coefficient. It is defined as follows:  
\[f(x_{k}+\alpha_{k}d_{k})-f(x_{k})\leq\delta\alpha_{k}g_{k}^{T}d_{k},\]0  
This parameter is similar to the PRP coefficient, which possesses convergence with both an exact line and inexact line searches, providing a sufficient descent condition. Many modifications of the WYL coefficient have been suggested. The reader can refer to [21]. Here we mention some of them. \[f(x_{k}+\alpha_{k}d_{k})-f(x_{k})\leq\delta\alpha_{k}g_{k}^{T}d_{k},\]1which were presented in [22; 23], respectively. [PERSON] et al. [24] proposed a new CG version called (G3 TCG) that offers many selections of CG parameters.