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, ZhejiangUniversity, Hangzhou, China)
(3Xi’anJiaotong-LiverpoolUniversity,Suzhou215123,China)
; ; †
Testing functions
All the test functions are minimization problems defined as following:
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 Miis the rotation matrix defined in the reference (Liang et al. 2013).
f1 Sphere
(1)
f2 Rotated High Conditioned Elliptic Function
(2)
f3 Rotated Bent Cigar Function
(3)
f4Rosenbrock's function
(4)
f5Griewank's function
(5)
f6Rastrigin's function
(6)
f7 Shifted and Rotated Rosenbrock’s Function
(7)
f8 Shifted and Rotated Weierstrass Function
(8)
f9 Shifted and Rotated Griewank’s Function
(9)
f10Schwefel's function
(10)
f11Weierstrass function
(11)
where a = 0.5, b=3, kmax=20.
f12 Shifted Rastrigin’s Function
(12)
f13 Shifted and Rotated Rastrigin’s Function
(13)
f14 Shifted Schwefel’s Function
(14)
f15 Shifted and Rotated Schwefel’s Function
(15)
f16 Shifted and Rotated HappyCat Function
(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 / fitnessf1 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. S2Convergence 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