A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 58

Table 2 Single-objective test functions
Name (comment) Formula: f ð x Þ Search domain

P i¼1 x i

[ - 100, 100] n Sphere (unimodal)

P i¼1 x jjþ Q i¼1 x jj

10, 10] n

Schwefel 2.22 (unimodal) [

-

2

P i ¼ 1 P j ¼ 1 x j

100, 100] n

Schwefel 1.2 (unimodal) [

-

h i

P n 1
i ¼ 1

2

[ - 30, 30] n

Rosenbrock (unimodal)

2

100 x i þ 1 x i þ x i 1 ð Þ

P i¼1 ix i þ

1.28, 1.28] n Noise (unimodal) [

random ½ 0 ; 1 Þ -

P i¼1

[ - 100, 100] n Step (unimodal)

2

ð b x i þ 0 : 5 c Þ

P i¼1 ð x i 10 cos ð 2 p x i Þþ 10 Þ

5.12, 5.12] n

Rastrigin (multimodal) [

-

p

4000 P i¼1 x i Q i¼1 cos

600, 600] n

Griewank (multimodal) [

-

x i

1

þ 1

q

1

n P i¼1 x i

32, 32] n Ackley (multimodal) [

-

20 þ e 20 exp 0 : 2

exp n P i¼1 cos

ð 2 p x i Þ

Table 3 The parameter settings of optimization algorithms
Algorithm

Parameter

GA (traditional crossover) P r ¼ 0 : 2 ; P c ¼ 0 : 4 ; P m ¼ 0 : 1 ; S ¼ 0 : 05, tournament method

for selection

GA (multiple-crossover) P r ¼ 0 : 2 ; P c ¼ 0 : 4 ; P m ¼ 0 : 1 ; S ¼ 0 : 05, tournament method

for selection

Standard PSO W ¼ 0 : 9 ; C 1 ¼ C 2 ¼ 2

The hybrid algorithm W 1 ¼ 0 : 9 ; W 2 ¼ 0 : 4 ; C 1i ¼ C 2f ¼ 2 : 5 ; C 1f ¼ C 2i ¼ 0 : 5 ;

f m ¼ 0 : 001 ; n tc ¼ n mc ¼ 0 : 2

Table 4 The comparison results among single-objective optimization algorithms for the Sphere
function

GA (traditional
crossover)

GA (multiple-
crossover)

PSO
(standard)

The hybrid
algorithm

10 − 14 10 − 18 10 − 98 10 − 119

Mean 3.28 × 2.56 × 2.25 × 1.58 ×

10 − 14 10 − 17 10 − 98 10 − 119 Standard

deviation

4.58 × 1.34 × 6.54 × 8.58 ×

algorithms. The population size and maximum iteration are set at 20 and 10,000,
accordingly.
By contrasting the results of GA with traditional crossover, GA with multiple-
crossover, standard PSO, and the hybrid of particle swarm optimization and the
genetic algorithm (Tables 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 and 12 ), it can be found that the
hybrid algorithm has a superior performance with respect to other optimization
algorithms. Moreover, the hybrid algorithm presents the best solutions in all test
functions except Schwefel 2.22, in which the PSO algorithm has the best solution