###### Abstract  
The conjugate gradient method is one of the most popular methods to solve large-scale unconstrained optimization problems since it does not require the second derivative, such as [PERSON]'s method or approximations. Moreover, the conjugate gradient method can be applied in many fields such as neural networks, image restoration, etc. Many complicated methods are proposed to solve these optimization functions in two or three terms. In this paper, we propose a simple, easy, efficient, and robust conjugate gradient method. The new method is constructed based on the Liu and Storey method to overcome the convergence problem and descent property. The new modified method satisfies the convergence properties and the sufficient descent condition under some assumptions. The numerical results show that the new method outperforms famous CG methods such as CG-Descent 5.3, Liu and Storey, and Dai and Liao. The numerical results include the number of iterations and CPU time. conjugate gradient; inexact line search; descent condition +
Footnote †: journal:  
0000-0001-9999-4000, 0000-0001-9999-4000  
**Inexact** line search, normally we use strong Wolfe-Powell (SWP) [1; 2], which is defined as follows:  
\[f(x_{k}+\alpha_{k}d_{k})\leq f(x_{k})+\delta\alpha_{k}g_{k}^{T}d_{k}, \tag{4}\]  
and  
\[\Big{|}g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\Big{|}\leq\sigma\,\Big{|}g_{k}^{T}d_{k }\Big{|}, \tag{5}\]  
where \(0<\delta<\sigma<1\). From Equation (4), we can note that  
\[f(x_{k}+\alpha_{k}d_{k})-f(x_{k})\leq\delta\alpha_{k}g_{k}^{T}d_{k},\]  
i.e.,  
\[f(x_{k+1})-f(x_{k})\leq\delta\alpha_{k}g_{k}^{T}d_{k}.\]  
If Equation (7) holds, then the following equation holds  
\[f(x_{k+1})\leq f(x_{k}).\]  
Also, note that Equation (5) can be written as follows  
\[\sigma\,g_{k}^{T}d_{k}\leq g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\leq-\sigma\,g_{k }^{T}d_{k}.\]  
Another version of Wolfe-Powell (WP) line search is the weak Wolfe-Powell (WWP) line search, which is defined by Equation (4) and the following equation  
\[g(x_{k}+\alpha_{k}d_{k})^{T}d_{k}\geq\sigma g_{k}^{T}d_{k}. \tag{6}\]  
The most famous classical formulas of the CG methods are [PERSON] (HS) [3], [PERSON] (PRP) [4], [PERSON] and Storey (LS) [5], [PERSON](FR) [6], [PERSON] (CD) [7], as well as [PERSON] and Yuan (DY) [8], given as follows:  
\[\beta_{k}^{HS}=\frac{g_{k}^{T}y_{k-1}}{d_{k-1}^{T}y_{k-1}},\,\,\beta_{k}^{PRP} =\frac{g_{k}^{T}y_{k-1}}{\|g_{k-1}\|^{2}},\,\,\beta_{k}^{LS}=-\frac{g_{k}^{T} y_{k-1}}{d_{k-1}^{T}\delta x_{k-1}}\]  
\[\beta_{k}^{FR}=\frac{\|g_{k}\|^{2}}{\|g_{k-1}\|^{2}},\,\,\beta_{k}^{CD}=-\frac {\|g_{k}\|^{2}}{d_{k-1}^{T}\delta x_{k-1}},\,\,\beta_{k}^{DY}=\frac{\|g_{k}\|^ {2}}{d_{k-1}^{T}\delta x_{k-1}},\]  
where \(y_{k-1}=g_{k}-g_{k-1}\). The first group of classical methods, i.e., (HS), (PRP), and (LS), are efficient. However, [PERSON] [9] proposed a counterexample to show that there exists a non-convex function, PRP, HS, and LS fail to satisfy the convergence properties even when the exact line search is used. [PERSON] suggested the importance to achieve the convergence properties of PRP, HS, and LS methods, where the methods should be non-negative. [PERSON] and [PERSON] [10] showed that non-negative PRP or HS, i.e., \(\beta_{k}=\max\{\beta_{k}^{HS},\,\,0\,\}\), is convergent under special different line searches. The second group of classical CG methods, i.e., (FR), (PRP), (DY), are robust and converge. However, the group is not efficient. The descent condition (downhill condition) performs an important ruling in the convergence of the CG method and its robustness, which is defined as follows  
\[g_{k}^{T}d_{k}<0.\]  
[PERSON] [11] proposed another version of the descent condition called the sufficient descent condition, which plays a significant role in the convergence of the CG method.