## 4 Numerical Results and Discussion  
To improve the efficiency of the proposed CG method, we modified Equation (16) as follows:  
\[\beta_{k}^{LS+}=\left\{\begin{array}{ll}-\frac{\left\|g_{k}\right\|^{2}- \left|g_{k}^{T}g_{k-1}\right|}{d_{k-1}^{2}g_{k-1}}&\mbox{ if }\left\|g_{k}\right\|^{2}>\left|g_{k}^{T}g_{k-1}\right|\\ -\frac{\left\|g_{k}\right\|^{2}-\mu_{k}\left|g_{k}^{T}g_{k-1}\right|}{d_{k-1} ^{2}g_{k-1}}&\mbox{ if }\left\|g_{k}\right\|^{2}>\mu_{k}\left|g_{k}^{T}g_{k-1}\right|\\ -\mu_{k}\frac{g_{k}^{T}g_{k-1}}{d_{k-1}^{2}g_{k-1}}&\mbox{ otherwise}\end{array}\right.\]  
To analyze the efficiency of the new formula, we selected several test problems in Appendix A from CUTEr [38]. We compared the CG-Descent, LS, DL+ coefficients based on the CPU time and the number of iterations. We employed the modified CG descent 6.8 [39] with the SWP line search with \(\delta=0.01\), \(\sigma=0.1\) and memory = 0 for LS, LS+, DL+ algorithms. To obtain the result of CG-Descent 5.3, we employ the CG-descent 6.3 with memory = 0. The norm of the gradient was employed as the stopping criterion, specifically \(\|g_{k}\|\leq 10^{-6}\) for all algorithms. The host computer is an AMD A4 and 4 GB of RAM. Theresults are shown in Figures 1 and 2. Note that a performance measure introduced by [PERSON] and [PERSON] [40] was employed. This performance measure was introduced to compare a set of solvers S on a set of problems \(\rho\). Assuming \(n_{s}\) solvers and \(n_{p}\) problems in S and \(\rho\), respectively, the measure \(t_{p,s}\) is defined as the computation time (e.g., the number of iterations or the CPU time) required for the solver \(s\) to solve the problem \(p\).  
To create a baseline for comparison, the performance of the solver \(s\) on the problem \(p\) is scaled by the best performance of any solver in S on the problem using the ratio  
\[r_{p,s}=\frac{t_{p,s}}{\min\{t_{p,s}:s\in S\}}\]  
Let the parameter \(r_{M}\geq r_{p,s}\) for all \(p\), \(s\) be selected. A further assumption made is that \(r_{p,s}=r_{M}\) if and only if the solver \(s\) does not solve the problem \(p\). As we would like to obtain an overall assessment of the performance of a solver, we defined the following measure  
\[P_{s}(t)=\frac{1}{n_{p}}\text{size}\{p\in\rho:r_{p,s}\leq t\}\]  
Thus, \(P_{s}(t)\) is the probability for a solver \(s\in S\) that the performance ratio \(r_{p,s}\) is within a factor \(t\in R\) of the best possible ratio. Suppose we define the function \(p_{s}\) as the cumulative distribution function for the performance ratio. In that case, the performance measure \(P_{s}:\;\mathbb{R}\rightarrow[0,1]\) for a solver is non-decreasing and piecewise continuous from the right. The value of \(p_{s}(1)\) is the probability that the solver achieves the best performance of all the solvers. In general, a solver with high values of \(P_{s}(t)\), which would appear in the upper right corner of the figure, is preferable.  
Figure 1: Performance measure based on the number of iterations.  
Figure 1 shows that LS+ outperforms CG-Descent 5.3, LS, and DL+ in the number of iterations. It is worth noting that LS fails to obtain the stationary point for some functions. Thus, it solves around 95% of test functions. According to Figure 2, which presents the CPU time, we note that the LS+ also outperforms CG-Descent, LS, and DL+. Also, we can note that CG-Descent outperforms DL+ and LS+ in both figures. Thus, the proposed method is more efficient than DL+ and CG-Descent and more efficient and robust than the original LS. Also, it is worth noting that the proposed method can be extended to a three term CG method by using a modified DL+ CG formula, which is our new paper in near future.