IEEE Computational Intelligence Magazine - February 2021 - 44

for the worst-case scenario search is 100, and the stopping criterion is 100 generations.
Compared with the test problems in the previous subsection, the new test problems are higher-dimensional than that in
the previous subsection, so the stopping criterion is extended
to 1000 real function evaluations. We run six compared algorithms (MMEA, SA-MMEA, MMDE, SA-MMDE, MMMFEA, and SA-MM-MFEA) on the new test problems for 30
independent times. The MSE of the obtained solutions to the

true optimum are shown in Table IV, where the results are
analyzed by the Friedman test with the Bergmann-Hommel
post-hoc test (SA-MM-MFEA is the control method and the
significance level is 0.05) [58].
Generally, SA-MM-MFEA performs the best on those
median-scale problems, SA-MMEA and SA-MMDE perform
worse than SA-MM-MFEA, and the other three compared
algorithms have the worst performance. As the dimension of
the decision space increases, the searching space dramatically

TABLE IV MSE of compared algorithms on new median-scale benchmark problems. The results are shown in the form of mean ±
standard deviation. The results are analyzed by the Friedman test with the Bergmann-Hommel post-hoc test (SA-MM-MFEA is the
control method and the significance level is 0.05). The best fitness values among all the compared algorithms for each problem are
highlighted.

44

MMEA

SA-MMEA

MMDE

SA-MMDE

MM-MFEA

SA-MM-MFEA

I-GRIEWANK

5.6 ± 1.0

1.2 ± 0.4

4.9 ± 1.2

1.2 ± 0.6

5.1 ± 0.7

0.0 ± 0.0

I-ACKLEY

7.9 ± 1.6

2.6 ± 1.8

6.9 ± 1.9

11.4 ± 3.0

6.0 ± 1.2

0.0 ± 0.0

I-RASTRIGIN

6.6 ± 1.3

1.0 ± 0.6

7.1 ± 1.5

1.2 ± 0.8

6.2 ± 1.2

0.0 ± 0.0

I-ELLIPSOID

5.8 ± 0.9

1.0 ± 0.4

5.5 ± 0.9

0.8 ± 0.4

5.7 ± 1.0

0.0 ± 0.0

I-ROSENBROCK

9.4 ± 2.3

4.7 ± 2.0

10.4 ± 1.3

4.4 ± 2.8

8.4 ± 2.1

4.9 ± 2.4

FRIEDMAN RANK

5.4

2.4

4.8

2.8

4.2

1.4

ADJUSTED P-VALUE

0.00

0.40

0.00

0.24

0.02

NA

II-GRIEWANK

5.4 ± 0.7

1.3 ± 0.4

4.9 ± 1.0

1.6 ± 1.0

5.1 ± 0.8

0.0 ± 0.0

II-ACKLEY

7.4 ± 1.4

4.0 ± 1.0

7.0 ± 1.6

13.3 ± 1.9

6.2 ± 1.5

0.1 ± 0.0

II-RASTRIGIN

6.6 ± 1.5

0.9 ± 0.5

7.1 ± 1.5

1.1 ± 0.5

6.6 ± 1.2

0.0 ± 0.0

II-ELLIPSOID

5.9 ± 1.1

1.0 ± 0.3

5.4 ± 1.1

0.7 ± 0.4

5.4 ± 0.8

0.0 ± 0.0

II-ROSENBROCK

9.7 ± 1.7

6.6 ± 2.1

12.0 ± 2.0

5.9 ± 1.8

9.5 ± 2.6

5.0 ± 2.0

FRIEDMAN RANK

5.4

2.4

4.8

3.2

4.2

1.0

ADJUSTED P-VALUE

0.00

0.24

0.00

0.06

0.01

NA

III-GRIEWANK

5.7 ± 1.1

1.3 ± 0.4

4.8 ± 0.8

1.4 ± 0.7

5.1 ± 0.8

0.0 ± 0.0

III-ACKLEY

6.2 ± 1.1

0.2 ± 0.1

7.0 ± 1.4

0.4 ± 0.3

5.9 ± 1.2

0.0 ± 0.0

III-RASTRIGIN

6.4 ± 1.5

0.8 ± 0.4

7.7 ± 1.4

1.3 ± 0.9

6.1 ± 0.9

0.0 ± 0.0

III-ELLIPSOID

5.8 ± 0.6

1.1 ± 0.4

5.1 ± 1.1

0.8 ± 0.4

5.5 ± 0.8

0.0 ± 0.0

III-ROSENBROCK

10.0 ± 2.6

5.5 ± 2.1

11.1 ± 1.4

6.5 ± 2.8

8.8 ± 2.2

6.8 ± 2.5

FRIEDMAN RANK

5.4

2.0

5.2

2.6

4.4

1.4

ADJUSTED P-VALUE

0.00

0.61

0.00

0.31

0.01

NA

IV-GRIEWANK

5.8 ± 1.0

1.2 ± 0.5

4.8 ± 0.8

1.3 ± 1.0

5.0 ± 0.6

0.0 ± 0.0

IV-ACKLEY

7.1 ± 0.9

4.4 ± 1.1

7.6 ± 2.1

14.0 ± 2.4

6.1 ± 1.3

0.4 ± 0.7

IV-RASTRIGIN

6.9 ± 1.0

0.8 ± 0.5

7.3 ± 1.6

1.4 ± 0.8

6.5 ± 1.0

0.0 ± 0.0

IV-ELLIPSOID

5.9 ± 1.2

1.0 ± 0.4

5.3 ± 0.9

0.8 ± 0.5

5.5 ± 0.8

0.0 ± 0.0

IV-ROSENBROCK

8.6 ± 2.0

4.5 ± 2.2

11.1 ± 1.7

3.9 ± 1.9

8.5 ± 2.8

3.5 ± 2.0

FRIEDMAN RANK

5.2

2.4

5.0

3.2

4.2

1.0

ADJUSTED P-VALUE

0.00

0.24

0.00

0.06

0.00

NA

V-GRIEWANK

5.5 ± 0.9

1.3 ± 0.4

5.0 ± 1.0

1.3 ± 0.9

5.3 ± 0.9

0.0 ± 0.0

V-ACKLEY

6.7 ± 1.2

0.0 ± 0.0

6.6 ± 1.5

0.1 ± 0.1

5.7 ± 1.5

0.0 ± 0.1

V-RASTRIGIN

6.6 ± 1.3

1.0 ± 0.6

7.2 ± 1.5

1.3 ± 0.6

6.4 ± 1.1

0.0 ± 0.0

V-ELLIPSOID

5.9 ± 0.9

0.9 ± 0.3

5.3 ± 0.8

0.9 ± 0.5

5.5 ± 0.9

0.0 ± 0.0

V-ROSENBROCK

9.8 ± 2.3

5.2 ± 2.0

10.7 ± 1.7

5.8 ± 2.5

9.0 ± 2.5

4.4 ± 2.4

FRIEDMAN RANK

5.6

2.6

5.0

2.4

4.4

1.0

ADJUSTED P-VALUE

0.00

0.18

0.00

0.24

0.00

NA

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2021
IEEE Computational Intelligence Magazine - February 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - February 2021
