## 5 Numerical Experiments  
In this section, we report some numerical results on some of the unconstrained optimization problems in the CUTE [21] test problem libraty. We test the ZPRP method and ZHS method, and compare the performance of the these two methods with the MPRP method in [18]. The parameters\(\delta~{}=~{}10^{-4},\rho=0.3\) and \(\mu=0.001\). All codes were written in MATLAB R2012a and run on PC with 3.00 GHz CPU processor and Win 7 operation system. We use the stopping iteration \(\|g_{k}\|_{\infty}\leq 10^{-6}\). The detailed numerical results are listed on the web site: [[http://mathxiuxiu.blog.sohu.com/326066259.html](http://mathxiuxiu.blog.sohu.com/326066259.html)]([http://mathxiuxiu.blog.sohu.com/326066259.html](http://mathxiuxiu.blog.sohu.com/326066259.html)).  
We first evaluate the performance of the ZPRP method with that of CG-DESCENT proposed by [PERSON] and [PERSON] (2005) and all methods with the Wolfe line search (6). Figures 1-3 show the numerical performance of the above methods related to the total number of iterations, the number of function and gradient evaluations, CPU time, respectively, which are evaluated using the profiles of [PERSON] and [PERSON] (2012). For each method, we plot the fraction \(P\) of problems for which the method is within a factor \(t\) of the smallest number of iterations, or the smallest number of function evaluations or least CPU time, respectively. The left side of the figure gives the percentage of the test problems for which a method is the fastest; the right side gives the percentage of the test problems that are successfully solved by each of the methods. The top curve is the method that solved the most problems in a time that was within a factor \(t\) of the best time. Clearly, the ZPRP method has the better performance since it illustrates the best probability of being the optimal solver, outperforming CG-DESCENT. From Figure 1, we can obtain the ZPRP method solves about 59.5% of the test problems with the least number of function evaluations while CG-DESCENT solve about 56.5% of the test problems. For the total number of function and gradient evaluations, in Figure 2 illustrates that the ZPRP method solves 55.2% of the test problems with the least number of function and gradient evaluations while CG-DESCENT solve about 52.2% of the test problems. Therefore, the ZPRP method outperforms CG-DESCENT.  
In the sequel, we compare the performance of the ZPRP method with that of the ZHS method and the MPRP method in [18] and all methods with the line search (9). Figures 4-6 list the performance of the above methods relative to CPU time, the number of function evaluations and the number of gradient evaluations, respectively. From Figure 4, we can observe that the ZPRP method outperforms the MPRP and ZHS method. More analytically, the performance profile for the number of iteration shows that ZPRP can solve 61% of the test problems with the least number while MPRP and ZHS solve about 47.5% and 45.2% of the test problems, respectively. As regards the number of function and gradient evaluations, Figure 5 shows that the ZPRP solves 80% of the test problems with the least number. Hence, the performance of the ZPRP method slightly better than that of the MPRP and ZHS methods.