## 3 Methodology
### The Well Sequence
The variant proposed WE-PSO, TO-PSO, and KN-PSO is compared with other initialized approaches Sobol sequence (SO-PSO), Halton Sequence (H-PSO), and standard PSO. The experimental results give superior results in higher dimensions for KN-PSO on other SO-PSO, H-PSO, PSO, and proposed approach TO-PSO and WE-PSO. Effect of Using Different Dimensions for Problems

The core objective of this simulation setup is to find the superiority of results depending upon the dimension of the functions that are to be optimized. In experiments, three dimensions for benchmark functions D = 10, D = 20, and D = 30 were used. Simulation results were presented in Table 2. From these simulation results, it was found that functions having larger dimensions found tougher to optimize and it can be seen from the Table 2 when dimension size is D = 20 and D = 30, and our proposed approach KN-PSO shows better result on higher dimensions than other approaches such as WE-PSO, TO-PSO, standard PSO, H-PSO, and SO-PSO. KN-PSO is compared with the other approaches such as WE-PSO, TO-PSO, SO-PSO, H-PSO and standard PSO, where each technique true value is presented for comparison with other techniques for the same nature of problems. Standard benchmark functions are presented in the Table 3, and their parameter settings are also shown in the Table 2. The Table 4 shows that with dimension D-30, KN-PSO is more superior and outperforms in convergence than the WE-PSO, TO-PSO, standard PSO, SO-PSO, and H-PSO. The comparative analysis can be seen from Table 4 that with a smaller dimension size, standard PSO performs well (D = 10), but as the size of dimension increases, KN-PSO outperforms in convergence significantly. Hence, KN-PSO is best for higher dimensions. The experimental results from Table 4 show that KN-PSO outclasses the results of WE-PSO, TO-PSO, SO-PSO, H-PSO, and traditional PSO for all functions. It can be seen that the TO-PSO outperformed the results of the other techniques in all functions on SO-PSO, H-PSO, and standard PSO, while the other approach H-PSO performed better on functions f4, f1, f2 for 20D. However,

Figure 22: Convergence curve on fn16. H-PSO gives an overall poor result on higher dimensions, and SO-PSO gives a slightly better result on functions f8, f9, f15 on 10-D but the worst result on larger dimensions). Standard PSO did not provide a better result. Figures from Figures 7-15 depict that WE-PSO outperforms in simulation results compared to other approaches for solving the standard benchmark tests functions for dim size D = 10, D = 20, and D = 30. For the validation of the numerical results, mean ranks obtained by [PERSON] and [PERSON] test for KN-PSO, WE-PSO, TO-PSO, SO-PSO, HA-PSO, and standard PSO are given in the Table 5.

### Discussion on DE Results

Population initialization is a vital factor in evolutionary computing-based algorithm, which considerably influences the diversity and convergence. Rather than applying the random distribution for initialization, quasirandom sequences are used to initialize the population. In this paper, the capability of DE has been extended to make it suitable for the optimization problem by introducing, new initialization techniques Knuth sequence-based DE (DE-KN), the Torus-based sequence-based (DE-TO) and the WELL sequence-based DE (DE-WE) to solve the optimization problems in large dimension search spaces. For global optimization, the most considerable variety of benchmark problems can be used. All benchmark problems have their own individual abilities, and the variety of detailed characteristics of such functions explains the level of complexity for benchmark problems. For the efficiency analysis of the abovementioned optimization algorithms, Table 3 displays the benchmark problems that are utilized. The Table 2 explains the following contents of benchmark problems: name, range, domain, and formulas.