## 2 The Proposed CG Formula and Its Motivation
Then, a constant \(\varepsilon>0\) exists such that  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]1  
Upon squaring both sides of (2), we obtain  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]2  
Dividing Equation (21) by \(\|g_{k}\|^{4}\) yields\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]3  
Using Equations (5) and (16), we now obtain  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]4  
Repeating the process for (22) using the relationship \(\frac{1}{\left\|g_{1}\right\|}=\frac{1}{\left\|d_{1}\right\|}\) yields  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]5  
From (20), we obtain  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]6  
Therefore,  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]7  
\(\square\)  
**Theorem 3**. : _Suppose the sequences \(\{x_{k}\}\) and \(\{d_{k}\}\) are obtained by using (1), (2), and \(\beta_{k}^{DL-HS}\), where \(\alpha_{k}\) is computed via the SWP line search (4) and (5). Then, the descent condition holds._  
Proof. : Let  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]8  
By multiplying (2) by \(g_{k}^{T}\), and substituting \(\beta_{k}^{DL-HS}\), we obtain  
\[=-\|g_{k}\|^{2}+\left(-\frac{\|g_{k}\|^{2}-\mu_{k}\|g_{k}^{T}g_{k-1 }\|}{d_{k-1}^{T}g_{k-1}}\right)g_{k}^{T}d_{k-1}. \tag{17}\]9  
\[\beta_{k}^{LS+}=\frac{-\|g_{k}\|^{2}+\mu_{k}\|g_{k}^{T}g_{k-1}}{d_{k-1}^{T}g_{ k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}+\frac{\mu_{k}|g_{k}^{T}g_{k-1 }|}{d_{k-1}^{T}g_{k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}=\frac{\| g_{k}\|^{2}}{\|g_{k-1}\|^{2}}\]0  
This completes the proof. \(\square\)  
The condition in (23) is called Zoutendijk condition [37], acting as an important role in proving convergence properties of the CG method. **Lemma 1**. : _Assume that Assumption 1 holds. Consider any form of (1) and (2) with step size \(\alpha_{k}\) satisfying the WWP line search, where the search direction \(d_{k}\) is descent. Then, we have_  
\[\beta_{k}^{LS+}=\frac{-\|g_{k}\|^{2}+\mu_{k}\|g_{k}^{T}g_{k-1}}{d_{k-1}^{T}g_{ k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}+\frac{\mu_{k}|g_{k}^{T}g_{k-1 }|}{d_{k-1}^{T}g_{k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}=\frac{\| g_{k}\|^{2}}{\|g_{k-1}\|^{2}}\]1  
Moreover, (23) holds for exact and SWP line searches. Substituting (7) into (18), it follows that  
\[\beta_{k}^{LS+}=\frac{-\|g_{k}\|^{2}+\mu_{k}\|g_{k}^{T}g_{k-1}}{d_{k-1}^{T}g_{ k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}+\frac{\mu_{k}|g_{k}^{T}g_{k-1 }|}{d_{k-1}^{T}g_{k-1}}\leq\frac{-\|g_{k}\|^{2}}{d_{k-1}^{T}g_{k-1}}=\frac{\| g_{k}\|^{2}}{\|g_{k-1}\|^{2}}\]2  
**Theorem 4**. : _Assume that Assumption 1 holds. Consider the conjugate gradient method in (1) and (2) with\(\beta_{k}^{DL-HS}\), where \(d_{k}\) is a descent direction and \(\alpha_{k}\) is obtained by the strong Wolfe line search. Then, the \(\liminf_{k\rightarrow\infty}\left\|g_{k}\right\|=0\)._  
**Proof. ** We will prove this theorem by contradiction. Suppose Theorem 4 is not true.