## 2 The New Smoothing Conjugate Gradient Method and its Global Convergence
We denote  
\[\tilde{x}_{k+1}=x_{k}+\tilde{\alpha}_{k}d_{k}\]  
\[\tilde{d}_{k+1}=-\
abla\Psi_{\mu_{k}}(\tilde{x}_{k+1})+\tilde{\beta}_{k+1}d_{k} \tag{9}\]  
By  
\[(\
abla\Psi_{\mu_{k}}(\tilde{x}_{k+1}))^{T}\tilde{d}_{k+1}=-\|\
abla\Psi_{\mu_{ k}}(\tilde{x}_{k+1})\|^{2}+\tilde{\beta}_{k+1}[(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d _{k}+\]  
\[(\
abla\Psi_{\mu_{k}}(\tilde{x}_{k+1})-\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k}] \tag{10}\]  
We know that \(\beta_{k}\) in Equation (3) is equivalent to (see [24])  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]0  
Since Assumption 1, Equations (10) and (11) yield  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]1  
by Mean Value Theorem, we have  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]2  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]3  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]4  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]5  
Then, we obtain that  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]6  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]7  
By Equations (12) and (13), we know that Equations (4) and (5) determine a positive stepsize \(\alpha_{k}\). And there must exists a constant \(\xi\in(0,1)\) yields  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]8  
Denote \(\omega=\min\{\frac{\xi}{L},\frac{2\xi(1-\delta)}{L}\}\), then Equation (7) holds. And Equation (5) implies that Equation (8) holds for \(k+1\). Hence, the proof is completed. **Theorem 1**. : Suppose that for any fixed \(\mu>0\), \(\Psi_{\mu}\) satisfies Assumption 1, then the infinite sequence \(\{x_{k}\}\) generated by Algorithm 1 satisfies  
\[\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k})\leq\Psi_{\mu_{k}}(x_{k})+\delta\alpha_{ k}(\
abla\Psi_{\mu_{k-1}}(x_{k}))^{T}d_{k} \tag{4}\]9  
**Proof** Denote \(K=\{k|\mu_{k+1}=m_{1}\mu_{k}\}\), we first show that \(K\) is an infinite set. If \(K\) is a finite set, there exists an integer \(\bar{k}\) such that  
\[(\
abla\Psi_{\mu_{k}}(x_{k}+\alpha_{k}d_{k}))^{T}d_{k+1}\leq-\sigma\|\
abla \Psi_{\  
In the following, we will give the analysis about the global convergence of Algorithm 1. (The Algorithm 2 is the algorithm framework of Algorithm 1.) Before doing this work, we need the following basic assumptions.  
**Assumption 1.**  
(i) For any \(x\in R^{n}\), \(0<\mu\leq\mu_{0}\), the level set \(L_{\mu}(c)=\{x\in R^{n}|\Psi_{\mu}(x)\leq c\}\) is bounded.  
(ii) \(\
abla\Psi_{\mu_{k}}(x_{k+1})\) is Lipschitz continuous, that is, there exists a constant \(L>0\) such that  
\[\|\
abla\Psi_{\mu_{k}}(x_{k+1})-\
abla\Psi_{\mu_{k-1}}(x_{k})\|\leq L\|x_{k+1} -x_{k}\|,\forall x_{k+1},x_{k}\in L_{\mu}(c)\]0  
for all \(k>\bar{k}\).