Supporting information

Advanced Discussion Mechanism based Brain Storm Optimization Algorithm

YutingYang1,2,YuhuiShi3,ShunrenXia†1,2

(1 KeyLaboratoryofBiomedicalEngineeringofMinistryofEducation,ZhejiangUniversity,
Hangzhou310027,China)

(2Zhejiang Provincial Key Laboratory of Cardio-Cerebral Vascular Detection Technology and Medicinal Effectiveness Appraisal, Zhejiang University, Hangzhou, China)

(3Xi’anJiaotong-LiverpoolUniversity,Suzhou215123,China)

; ; †

Testing functions

All the test functions are minimization problems defined as following:

Minfx,x=x1,x2,…,xDT

D is the dimensions.

The formulas and details of benchmark functions used to test our algorithm are presented below. oi is the shifted global optimum, and Fi*is the minimum, and Mi is the rotation matrix defined in the reference (Liang et al. 2013).

f1 Sphere

f1x=i=1Dxi2 (1)

f2 Rotated High Conditioned Elliptic Function

f2x=i=1D106i-1D-1zi2+F1* z=Mx-o1 (2)

f3 Rotated Bent Cigar Function

f3x=z12+106i=2Dzi2+F2* z=Mx-o2 (3)

f4 Rosenbrock's function

f4x=i=1D-1100xi2-xi+12+xi-12 (4)

f5 Griewank's function

f5x=i=1Dxi24000-cosxii+1 (5)

f6 Rastrigin's function

f6x=i=1Dxi2-10cos2πxi+10 (6)

f7 Shifted and Rotated Rosenbrock’s Function

f7x=f4M2.048x-o4100+1+F4* (7)

f8 Shifted and Rotated Weierstrass Function

f8x=f11M0.5x-o6100+F6* (8)

f9 Shifted and Rotated Griewank’s Function

f9x=f5M600x-o7100+F7* (9)

f10 Schwefel's function

f10x=418.9829×D-i=1Dxi'sinxi'12 xi'=xi+4.209687462275036e+002 (10)

f11 Weierstrass function

f11x=i=1Dk=0kmaxakcos2πbkxi+0.5-Dk=0kmaxakcos2πbk∙0.5 (11)

where a = 0.5, b=3, kmax=20.

f12 Shifted Rastrigin’s Function

f12x=f65.12x-o8100+F7* (12)

f13 Shifted and Rotated Rastrigin’s Function

f13x=f6M5.12x-o9100+F9* (13)

f14 Shifted Schwefel’s Function

f14x=f101000x-o10100+F10* (14)

f15 Shifted and Rotated Schwefel’s Function

f15x=f10M1000x-o11100+F11* (15)

f16 Shifted and Rotated HappyCat Function

f16x=i=1Dzi2-D14+0.5i=1Dzi2+i=1DziD+0.5+F13*z=M5x-o13100 (16)

f17 Composition function 1 (CF1) in (Liang et al. 2005): CF1 is composed using ten sphere functions.

f18 Composition function 5 (CF5) in (Liang et al. 2005): CF5 is composed using ten different benchmark functions, whose global optimum is even more difficult than CF1 to locate.

The global fitness values and search ranges, [Xmin, Xmax], are given in Table 1. The initial ranges of each function are set the same as the search ranges.

Table 1 global optimum and search ranges of the test functions

BFs / search range / fitness
f1 sphere function / [-100,100]D / 0
f2 Rotated High Conditioned Elliptic Function / [-100,100]D / 100
f3 Rotated Bent Cigar Function / [-100,100]D / 200
f4Rosenbrock's function / [-2.048, 2.048]D / 0
f5Griewank's function / [-600,600]D / 0
f6Rastrigin's function / [-5.12, -5.12]D / 0
f7 Shifted and Rotated Rosenbrock’s Function / [-100,100]D / 400
f8 Shifted and Rotated Weierstrass Function / [-100,100]D / 600
f9 Shifted and Rotated Griewank’s Function / [-100,100]D / 700
f10Schwefel's function / [-500,500]D / 0
f11Weierstrass function / [-0.5,0.5]D / 0
f12 Shifted Rastrigin’s Function / [-100,100]D / 800
f13 Shifted and Rotated Rastrigin’s Function / [-100,100]D / 900
f14 Shifted Schwefel’s Function / [-100,100]D / 1000
f15 Shifted and Rotated Schwefel’s Function / [-100,100]D / 1100
f16 Shifted and Rotated HappyCat Function / [-100,100]D / 1300
f17 Composition Function 1 (CF1) / [-5,5]D / 0
f18 Composition Function 5 (CF5) / [-5,5]D / 0

Experiment results

Convergence progresses of each algorithm on different benchmark functions for both 10-D and 30-D are shown in the following figures.

f2 f3

f4 f5

f7 f8

f9 f10

f11 f12

f14 f15

f16 f18

Fig. S1Convergence progresses of different algorithms for 10-D problems

f2 f3

f4 f5

f7 f8

f9 f10

f11 f12

f14 f15

f16 f18

Fig. S2 Convergence progresses of different algorithms for 30-D problems

References

Liang J, Qu B, Suganthan P (2013) Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization. Computational Intelligence Laboratory

Liang JJ, Suganthan PN, Deb K (2005) Novel composition test functions for numerical global optimization. In: Swarm Intelligence Symposium (SIS), Pasadena, CA, 2005. IEEE, pp 68-75. doi:10.1109/SIS.2005.1501604