We have used two ill-posed functions to illustrate the optimization characteristics of the DFPSO algorithm compared with

the PSO algorithm and the GPSO algorithm。 Hence, the PSO algorithm and the GPSO algorithm are still used as comparison

Fig。 9。 Success rates and average iterations for the Griewank function for PSO, GPSO, and DFPSO。

algorithms in this subsection。 Moreover, we add the second class algorithms (SPSO2007, SPSO2011, CLPSO, and ALC-PSO) into the comparison algorithms。 The SPSO2007 algorithm, the SPSO2011 algorithm, the CLPSO algorithm, and the ALC- PSO algorithm are also well-performed and popular variants among the PSO algorithms where the impacts of communica- tion topologies on the optimization performance are   elaborated。

Specifically, Bratton and Kennedy [5] defined a standard particle swarm optimization algorithm (SPSO2007) by summa- rizing the various variants of the PSO algorithm and research experiences, and suggested that the SPSO2007 algorithm

Fig。 10。 Success rates and average iterations for the Rastrigin function for PSO, GPSO, and DFPSO。

should be used as a standard algorithm of evaluating the performance capabilities of the improved PSO algorithms。 In 2013, Zambrano-Bigiarini et al。 [46] (2013) published a new version of the SPSO algorithm (SPSO2011) based on the recent theo- retical developments including the adaptive random topology and rotational invariance, and suggested that the SPSO2011 algorithm should also be used as a standard algorithm of evaluating the performance capabilities of the improved PSO algo- rithms。 Hence, the SPSO2007 algorithm and the SPSO2011 algorithm as important nodes in the PSO algorithms’ development process are used as comparison algorithms。 Furthermore, the CLPSO algorithm, which has the higher citations and impacts in

Table 1

Test  functions。

Functions n Optimum Domain Name

Pn      2

f1 ðxÞ ¼ 

i¼1 xi 30 0 [—100, 100] Sphere

Pn Qn

f2 ðxÞ ¼ 

i¼1 jxi jþ 

i¼1 jxi j 30 0 [—10, 10] Schwefel’s P2。22

Pn      Pi

2 30 0 [—100, 100] Schwefel’s P1。2

f3 ðxÞ ¼ 

i¼1 ð

j¼1 xj Þ

Pn—1

2   2 2

30 0 [—2, 2] Rosenbrock

f4 ðxÞ ¼     i¼1  ½100ðxiþ1  — xi  Þ   þ ðxi  — 1Þ  ]

Pn 2 30 0 [—100, 100] Step

f5 ðxÞ ¼   i¼1 ðbxi þ 0:5cÞ

Pn      2

f6 ðxÞ ¼ 

i¼1 zi  þ f  bias1 ; z ¼ x — o