f22 ðxÞ ¼ 

i¼1 ½zi   — 10 cosð2pzi Þþ 10]þ f  bias9 30 —330 [—5,  5] Shifted  Rastrigin

z ¼ x — o

Pn 2

f23 ðxÞ ¼ 

i¼1 ½zi   — 10 cosð2p

z ¼ ðx — oÞ× M

Pn       P20 k k

20 k k

30 90 [—0。5, 0。5] Shifted  Rotated Weierstrass

f24 ðxÞ ¼ 

i¼1 ð

k¼0 ½0:5   cosð2p3  ðxi  þ 0:5ÞÞ]Þ — nP

½0:5   cosðp3  Þ] þ f  bias11

z ¼ ðx — oÞ× M

Pn 2

30 —460 ½—p; p] Schwefel’s   P2。13

f25 ðxÞ ¼ 

Ai  ¼ Pn

i¼1 ðAi — Bi ðxÞÞ  þ f bias12

ðaij sin aj þ bij cos aj Þ

i¼1

B Pn

i¼1

ðaij sin xj þ bij cos xj Þ

8 kðxj — aÞ ; xj  > a

1  ui ðxj ; a; k; mÞ ¼ < 0; —a 6 xj  6 a 。

: kð—xj — 1 m ;   xj < —a

the field of the PSO algorithms, is one of the simple and well-performed algorithms for the multimodal functions and can be regarded as a representative algorithm from 2001 to 2007。 Finally, the ALC-PSO algorithm is also a well-performed algorithm published in 2013 in one of top journals。 Thus, the ALC-PSO algorithm is chosen as a comparison algorithm。

Moreover, the parameters of the comparison algorithms can be found in the corresponding references。 For the DFPSO algorithm, we set x ¼ —0:7 if the test functions are the unimodal functions or x ¼ —2:1 if the test functions are the multi- modal functions。 Other parameters are Dt ¼ 0:5; c ¼ 0:5; a ¼ 8:5, and b ¼ 0:01。 The population size is set as 20 for all algorithms。

Remark 12。 It is worth noting that we use (3) to calculate piðkÞ and the method proposed by the SPSO2011 algorithm [33] to compute piðk þ DkÞ for twenty-five benchmark functions。

4。2。1。Numerical  results  on  unimodal functions

In Table 2, the mean function errors, the minimal function errors, and the standard errors given by the different PSO vari-

5

ants after 1 × 10

function evaluations are reported。 One can see from this table that the results yielded by the DFPSO   algo-

rithm are better than those obtained by the other algorithms for a majority of test functions。 Moreover, the statistical results are also reported by using a two-sample Wilcoxon rank sum test。 The Wilcoxon rank sum test as a nonparametric statistical test can be used to determine the statistical significance of the difference between two independent algorithms。 For most of the test functions, the difference between the results yielded by the DFPSO algorithm and the ones obtained by the other algorithms are statistically significant。 Finally, the number of functions where the DFPSO algorithm yields significantly better or worse results is also shown in Table  2。