## 2 The Modified PRP Method and Its Properties  
In order to have the sufficient descent condition and keep simple structure and good properties, we take a modification to the denominator of the PRP formula, namely,  
\[\beta_{k}^{ZPRP}=\frac{g_{k}^{T}y_{k-1}}{max\{\mu\|d_{k-1}\|\,\|y_{k-1}\|,\|g _{k-1}\|^{2}\}}, \tag{13}\]  
where \(\mu>0\). For convenience, we call the iterative form by (2), (11) and (13) a ZPRP method. It is obvious that the ZPRP method reduces to the PRP method if \(\mu\|d_{k-1}\|\,\|y_{k-1}\|\leq\|g_{k-1}\|^{2}\). Then, we give the modified PRP type conjugate gradient method below (Algorithm 1). The following lemma shows that the direction \(d_{k}\) determined by (11) satisfies a trust region property. Lemma 1: _Let \(d_{k}\) be defined by (11) with \(\beta_{k}^{ZPRP}\), then we have_  
\[\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|. \tag{14}\]  
Proof of Lemma 1.: By (13), for all \(k\geq 1\), we have  
\[|\beta_{k}^{ZPRP}| \leq \frac{\|g_{k}\|\ \|y_{k-1}\|}{max\{\mu\|d_{k-1}\|\ \|y_{k-1}\|,\|g_{k-1}\|^{2}\}} \tag{15}\] \[\leq \frac{\|g_{k}\|}{\mu\|d_{k-1}\|}\]  
From (11), (13) and (15), we obtain  
\[\|d_{k}\| \leq \|g_{k}\|+|\beta_{k}^{ZPRP}|\ \|d_{k-1}\|+|\beta_{k}^{ZPRP}|\frac{\|g_{k} \|\|d_{k-1}\|}{|g_{k}^{T}y_{k-1}|}\|y_{k-1}\| \tag{16}\] \[\leq \|g_{k}\|+\frac{1}{\mu}\|g_{k}\|+\frac{|g_{k}^{T}y_{k-1}|}{\mu\|d _{k-1}\|\ \|y_{k-1}\|}\frac{\|g_{k}\|\ \|d_{k-1}\|}{|g_{k}^{T}y_{k-1}|}\|y_{k-1}\|\] \[= (1+\frac{2}{\mu})\|g_{k}\|.\]  
The proof is completed. **3. Global Convergence of the ZPRP Method**  
In this section, we come to show the global convergence of our proposed method. The following assumptions are often used in the literature to analyze the global convergence of conjugate gradient methods with inexact line searches. **Assumption 1**  
(i) The level set \(\Omega=\{x\in R^{n}|f(x)\leq f(x_{0})\}\) is bounded. (ii) In some neighborhood \(N\) of \(\Omega\), \(f\) is continuously differentiable and its gradient is Lipschitz continuous, that is, there exists a constant \(L>0\) such that \(\|g(x)-g(y)\|\leq L\|x-y\|,\forall x,y\in N\). We first prove the ZPRP method is globally convergent with [PERSON] line search (6). Under Assumption 1, we give a useful Zoutendijk condition [20]. Lemma 2: _Suppose that Assumption 1 holds. Consider the method in the form of (2) and (3) where \(d_{k}\) is a descent direction and \(\alpha_{k}\) satisfies the Wolfe line search conditions (6). Then we have_  
\[\sum_{k\geq 0}\frac{(g_{k}^{T}d_{k})^{2}}{\|d_{k}\|^{2}}<+\infty. \tag{17}\]  
Obviously, the Zoutendijk condition (17) and (12) imply that \[\sum_{k\geq 0}\frac{\|g_{k}\|^{4}}{\|d_{k}\|^{2}}<+\infty. \tag{18}\]  
**Theorem 1**. : _Suppose that Assumption 1 holds. Consider the ZPRP method, and \(\alpha_{k}\) is obtained by the Wolfe conditions (6). Then we have_  
\[\lim_{k\rightarrow\infty}\|g_{k}\|=0. \tag{19}\]  
**Proof of Theorem 1. ** By Lemma 1, we have \(\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|\). Let \(C=1+\frac{2}{\mu}\), then we get  
\[\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|. \tag{14}\]0  
which implies  
\[\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|. \tag{14}\]1  
Hence, (19) holds. The proof is completed. Next, we prove the global convergence of the ZPRP method under the condition (9). **Theorem 2**. : _Suppose that Assumption 1 holds. Consider the ZPRP method and \(\alpha_{k}\) satisfies the Armijo line search (9). Then we have_  
\[\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|. \tag{14}\]2  
**Proof of Theorem 2. ** Suppose that the conclusion is not true. Then there exists a constant \(\epsilon>0\) such that \(\forall k\geq 0\),  
\[\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|. \tag{14}\]3  
From (9) and Assumption 1 (i), we have  
\[\|d_{k}\|\leq(1+\frac{2}{\mu})\|g_{k}\|. \tag{14}\]4  
If \(\liminf_{k\rightarrow\infty}\alpha_{k}>0\), we get from (24) that \(\liminf_{k\rightarrow\infty}\|d_{k}\|=0\). From (12), we get \(\liminf_{k\rightarrow\infty}\|g_{k}\|=0\), which contradicts (23).