## 6 Results and Discussion
KHA performing better than MA. This convergence performance is almost exactly repeated with the two four-variable functions, the Powell function (Figure 6g) and with the Wood function (Figure 6h). Figure 6: _Cont._  
The evolution of the mean best values of the three algorithms for the five-variable functions, which are the Extended Cube, Shekel and Sphere functions, are reported in Figure 7a-c, respectively. For the Extended Cube function, MAKHA was able to obtain the global minimum, while the other two methods failed to converge to the global minimum. For the Shekel function, MAKHA obtained the global minimum very efficiently, as compared to MA, which obtained it at a higher NFE. KHA failed to converge to the global minimum for this function. A relative close pattern is repeated with the five-variable sphere function. Hartman function is a fifty-variable function and its results are depicted in Figure 7d. MAKHA and KHA converged to the global minimum efficiently, while MA failed to achieve it after 300,000 function evaluations. For the Griewank function, shown in Figure 7e, the three algorithms failed to converge to the global minimum. However, MAKHA performance was considerable better than the performance of the other two algorithms, which performed similarly. The Rastrigin function is one of the three functions in which MAKHA was not the top performer. The results for the Rastrigin function are reported in Figure 7f. The three methods did not obtain the global minimum. However, MA performed better than MAKHA, which in turn performed better than KHA. For the Rosenbrock function, whose results are shown in Figure 7g, the three algorithms did not obtain the global minimum, within the acceptable tolerance, with 1,500,000 function evaluations. However, the best value obtained by MAKHA is seven orders-of-magnitude better than that obtained by the other two algorithms. Sine Envelope Sine function is the second function in which MAKHA did not outperform MA, see Figure 7h. MA was the only method to achieve the global minimum. MAKHA's performance was significantly better than KHA's performance. Figure 6: Evolution of mean best values for MA, KHA and MAKHA for: (**a**) Goldstein–Price (**b**) [PERSON], (**c**) Levy 13, (**d**) Schaffer functions, (**e**) [PERSON], (**f**) Helical Valley, (**g**) [PERSON] and (**h**) [PERSON]. Figure 7: Evolution of mean best values for MA, KHA and MAKHA for: (**a**) Extended cube, (**b**) [PERSON], (**c**) Sphere, (**d**) [PERSON], (**e**) [PERSON] (**f**) [PERSON], (**g**) Rosenbrock, and (**h**) Sine Envelope Sine functions. Figure 8a-c shows the results of the mean best value obtained by the three algorithms for Styblinski-Tang, Trigonometric, and Zacharov functions, respectively. MAKHA was the only method to obtain the global minimum for Styblinski-Tang function and it did efficiently as measured by NFE. MA outperformed KHA for this particular function. The same pattern was obtained with the Trigonometric function as depicted in Figure 8b. The Zacharov function is the third function, which MAKHA did not outperform MA, as shown in Figure 8c. Table 4 shows a summary of the performance results for the twenty-seven benchmark problems. MAKHA has outperformed its parent algorithms in the majority of the benchmark problems studied. Best global values are shown in bold. The performance profiles reported in Figure 9 summarize the results of the MAKHA evaluation in comparison with the two original algorithms. MAKHA was the best algorithm in 24 out of the 27 cases considered. In several cases, the hybrid algorithm was the only  
Figure 8: Evolution of mean best values for MA, KHA and MAKHA for: (**a**) Styblinski-Tang, (**b**) Trigonometric and (**c**) Zacharov functions. algorithm that obtained the global minimum. Also, due to its efficient exploration and exploitation components, it converges to the global minimum with less NFE than the original two algorithms.