## 6 Results and Discussion  
As stated, each of the numerical experiments was repeated 30 times with different random seeds for MAKHA and the original MA and KHA algorithms. Parameters used for stochastic algorithms are reported in Table 3. The objective function value at each iteration for each trial was recorded. The mean and the standard deviation of the function values were calculated at each iteration. The global optimum was considered to be obtained by the method if it finds a solution within a tolerance value of \(10^{-10}\). The progress of the mean values is presented in Figures 5-8 for each benchmark function and a brief discussion of those results follows. The Ackley function has one minimum only. The global optimum was obtained using MAKHA and in relatively small number of function evaluations as shown in Figure 5a. MA and KHA were not able to obtain satisfactorily the global minimum. The best value obtained by MA was still improving by the end of the run, whereas KHA results were not. This significant improvement in performance was also clear with the Beale function (Figure 5b). The Beale function has one minimum only, which was only obtained satisfactory by MAKHA. The performance pattern for the three methods was different for the Bird function, as depicted in Figure 5c. MAKHA arrived at the global minimum almost instantaneously. KHA was trapped in a local minimum, while MA was approaching the global minimum, but did not reach it after 300,000 function evaluations. For the Booth function, both MAKHA and KHA arrived at the global minimum but with an improvement in orders of magnitude for the efficiency of MAKHA. On the other hand, MA failed to arrive at the global minimum, as shown in Figure 5d. For Bukin 6 function, none of the three methods obtained the global minimum within the used tolerance. However, MAKHA performed better than KHA, which performed better than MA, as shown in Figure 5e. MAKHA and KHA performed similarly for the Carrom table function, as they obtained the global minimum almost instantaneously, as shown in Figure 5f. MA could not achieve the global minimum even after 300,000 function evaluations. For the Cross-leg table function, MAKHA was the only method to obtain the global minimum, as shown in Figure 5g. Note that MA performed better than KHA after many NFE. MAKHA obtained the global minimum for the Generalized Eggholder function almost instantaneously, as depicted in Figure 5h. The other two methods were not able to obtain the global minimum but MA's performance was significantly better than KHA's. Figure 5: _Cont._  
For the Goldstein-Price function, all three algorithms obtained the global minimum, as depicted in Figure 6a. However, MAKHA and KHA were orders-of-magnitude more efficient than MA. For the Himmelblau function, MAKHA obtained the global minimum at low NFE, KHA obtained it at high NFE, and MA was not able to obtain it after 300,000 NFE, as shown in Figure 6b. This performance was almost exactly repeated with the Levy 13 function, as shown in Figure 6c. As for the Schaffer function, both MAKHA and MA converged to the global minimum within the used tolerance. MAKHA was more efficient than MA in terms of the NFE required to obtain the global minimum, as shown in Figure 6d. KHA failed to converge to the global minimum for this particular function. Figure 6e shows the evolution pattern for the Zettl function. MAKHA and KHA obtained the global minimum within a small NFE, while MA obtained it after considerably more NFE. The Helical Valley function has three variables. The evolution of the mean best values of the three algorithms is reported in Figure 6f and results showed that the performance of MAKHA was better than the other two algorithms. MAKHA efficiently obtained the global minimum, while the other two could not, with  
Figure 5: Evolution of mean best values for MA, KHA and MAKHA for: (**a**) Ackley, (**b**) Beale, (**c**) Bird (**d**) Booth, (**e**) Bukin6 (**f**) Carrom table, (**g**) Cross-leg table, and (**h**) Generalized Eggholder functions.