## 2 The Proposed CG Formula and Its Motivation  
The CG method with \(\beta_{k}^{LS}\) cannot satisfy the decent condition. On the other hand, \(\beta_{k}^{LS}\) comes from the group of efficient CG method as explained before to inherit the efficiency of \(\beta_{k}^{LS}\). To avoid the convergence problem, we use \(\beta_{k}^{AZPRP}\) in [19] and \(\beta_{k}^{DL}\) in [12] to propose a new non-negative CG method that satisfies the sufficient descent condition with SWP line search. Moreover, convergence properties are achieved. The new formula is a modification of \(\beta_{k}^{LS}\),\(\beta_{k}^{DL}\), and \(\beta_{k}^{AZPRP}\) defined as follows:  
\[\beta_{k}^{LS+}=\left\{\begin{array}{ll}-\frac{\|g_{k}\|^{2}-\mu_{k}|g_{k}^ {T}g_{k-1}^{T}}{d_{k-1}^{T}g_{k-1}}\ \text{if}\ \|g_{k}\|^{2}>\mu_{k}|g_{k}^{T}g_{k-1}|,\\ \beta_{k}^{DL-HS}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
\[g_{k}^{T}d_{k} =g_{k}^{T}(-g_{k}+\beta_{k}d_{k-1})=-\|g_{k}\|^{2}+\beta_{k}g_{k}^{T}d _{k-1}\] \[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]  
By using the proof by induction technique, for \(k=1\), it is true, i.e., \(g_{1}^{T}d_{1}=-\|g_{1}\|^{2}\). We now assume that it is true until \(k-1\), thus we can write Equation (16) as follows:  
\[\beta_{k}^{LS+}=\frac{-\|g_{k}\|^{2}+\mu_{k}\|g_{k}^{T}g_{k-1}}{d_{k-1}^{T}g_{ k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}+\frac{\mu_{k}|g_{k}^{T}g_{k-1 }|}{d_{k-1}^{T}g_{k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}=\frac{\| g_{k}\|^{2}}{\|g_{k-1}\|^{2}}\]  
Divide both sides of Equation (18) by \(\|g_{k}\|^{2}\) and using (5), we obtain  
\[-1+\sigma\frac{g_{k-1}^{T}d_{k-1}}{\left|g_{k-1}\right|^{2}}\leq\frac{g_{k}^{ T}d_{k}}{\left|g_{k}\right|^{2}}\leq-1-\sigma\frac{g_{k-1}^{T}d_{k-1}}{ \left|g_{k-1}\right|^{2}}. \tag{18}\]  
Repeating the process for the left-hand side of Equation (18), we obtain  
\[-1+\sum_{j=1}^{k-1}\sigma^{j}=-\sum_{j=0}^{k-1}\sigma^{j}.\]  
Repeating the process for the right hand side of Equation (18) yields  
\[-1-\sum_{j=1}^{k-1}\sigma^{j}=-2+\sum_{j=0}^{k-1}\sigma^{j}.\]  
Thus, Equation (18) can be written as follows:  
\[-\sum_{j=0}^{k-1}\sigma^{j}\leq\frac{g_{k}^{T}d_{k}}{\left|g_{k}\right|^{2}} \leq-2+\sum_{j=0}^{k-1}\sigma^{j}\]  
Since  
\[\sum_{j=0}^{k-1}(\sigma)^{j}<\frac{1-(\sigma)^{k}}{1-\sigma},\]  
hence  
\[-\frac{1-(\sigma)^{k}}{1-\sigma}\leq\frac{g_{k}^{T}d_{k}}{||g_{k}||^{2}}\leq- 2+\frac{1-(\sigma)^{k}}{1-\sigma}.\]  
When \(\sigma\leq 1/2\), we obtain \(\frac{1-(\sigma)^{k}}{1-\sigma}<2\). Let \(c=2-\frac{1-(\sigma)^{k}}{1-\sigma}\), then  
\[c-2\leq\frac{g_{k}^{T}d_{k}}{||g_{k}||^{2}}\leq-c. \tag{19}\]  
The proof is complete. \(\square\)  
The Convergence Property of the CG Method with \(\beta_{k}^{LS+}\) to Obtain the Stationary Point  
**Theorem 2**. : _Let Assumption 1 hold. Consider any form of (1) and (2), with the new formula (10), in which \(\alpha_{k}\) is obtained from the SWP line search (4) and (5) with \(\leq 1/2\). Then,_  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]0  
**Proof. ** We will prove the theorem by contradiction. Assume that the conclusion is not true.