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Table 1 Benchmark functions, D Dimension, C Characteristic, U Unimodal, M Multimodal, S Separable, N Non-separable

From: A teaching learning based optimization based on orthogonal design for solving global optimization problems

No. Function D C Range Formulation Value
f 1 Step 30 US [−100,100] f x = i = 1 D x i + 0.5 2 f min =0
f 2 Sphere 30 US [−100,100] f x = i = 1 D x i 2 f min =0
f 3 SumSquares 30 US [−100,100] f x = i = 1 D i x i 2 f min =0
f 4 Quartic 30 US [−1.28,1.28] f x = i = 1 D i x i 4 + random 0 , 1 f min =0
f 5 Zakharov 10 UN [−5,10] f x = i = 1 D x i 2 + i = 1 D 0.5 i x i 2 + i = 1 D 0.5 i x i 4 f min =0
f 6 Schwefel 1.2 30 UN [−100,100] f x = i = 1 D j = 1 i x j 2 f min =0
f 7 Schwefel 2.22 30 UN [−10,10] f x = i = 1 D x i + i = 1 D x i f min =0
f 8 Schwefel 2.21 30   [−100,100] f x = max i x i , 1 i D f min =0
f 9 Bohachevsky1 2 MS [−100,100] f x = x 1 2 + 2 x 2 2 0.3 cos 3 π x 1 0.4 cos 4 π x 2 + 0.7 f min =0
f 10 Bohachevsky2 2 MS [−100,100] f x = x 1 2 + 2 x 2 2 0.3 cos 3 π x 1 * cos 4 π x 2 + 0.3 f min =0
f 11 Bohachevsky3 2 MS [−100,100] f x = x 1 2 + 2 x 2 2 0.3 cos ( 3 π x 1 + 4 π x 2 ) + 0.3 f min =0
f 12 Booth 2 MS [−10,10] f(x) = (x1 + 2x2 − 7)2 + (2x1 + x2 − 5)2 f min =0
f 13 Rastrigin 30 MS [−5.12,5.12] f x = i = 1 D x i 2 10 cos 2 π x i + 10 f min =0
f 14 Schaffer 2 MN [−100,100] f x = si n 2 x 1 2 + x 2 2 0.5 1 + 0.001 x 1 2 + x 2 2 2 f min =0
f 15 Six hump camel back 2 MN [−5,5] f x = 4 x 1 2 2.1 x 1 4 + 1 3 x 1 6 + x 1 x 2 4 x 2 2 + 4 x 2 4 f min = − 1.03163
f 16 Griewank 30 MN [−600,600] f x = 1 4000 i = 1 D x i 2 i = 1 D cos x i i + 1 f min =0
f 17 Ackley 30 MN [−32,32] f x = 20 exp 0.2 1 D i = 1 D x i 2 exp 1 n i = 1 D cos 2 * pi * x i + 20 + e f min =0
f 18 Multimod 30 MS [−10,10] f x = i = 1 ` D x i i = 1 D x i f min =0
f 19 Noncontinuous rastrigin 30 MS [−5.12,5.12] f x = i = 1 D y i 2 10 cos 2 π y i + 10 Where y i = x i x i < 0.5 round 2 x i 2 x i 0.5 f min =0
f 20 Weierstrass 30 MS [−0.5, 0.5] f x = i = 1 D k = 0 kmax a k cos 2 π b k x i + 0.5 D k = 0 kmax a k cos 2 π b k x i + 0.5 , where a = 0.5 , b = 3 , kmax = 20 f min =0