Table 1: Standard benchmark functions adopted in this work.

Function Problem Range 𝑓 ( 𝑥 ) Classification

Sphere 𝑛 𝑖 = 1 𝑥 2 𝑖 [ 1 0 0 ; 1 0 0 ] 0 Unimodal
Rastrigin 𝑛 𝑖 = 1 ( 𝑥 2 𝑖 1 0 c o s ( 2 𝜋 𝑥 𝑖 ) + 1 0 ) [ 5 . 1 2 ; 5 . 1 2 ] 0 Multimodal
Griewank ( 1 / 4 0 0 0 ) 𝑛 𝑖 = 1 𝑥 2 𝑖 𝑛 𝑖 = 1 c o s ( 𝑥 𝑖 / 𝑖 ) + 1 [ 6 0 0 ; 6 0 0 ] 0 Multimodal
Rosenbrock 𝑛 1 𝑖 = 1 ( 1 0 0 ( 𝑥 𝑖 + 1 𝑥 2 𝑖 ) 2 + ( 𝑥 𝑖 1 ) 2 ) [ 2 . 0 4 8 ; 2 . 0 4 8 ] 0 Unimodal
Quartic ( 𝑛 𝑖 = 1 𝑖 𝑥 4 𝑖 ) + r a n d [ 0 , 1 ] [ 1 . 2 8 ; 1 . 2 8 ] 0 Noisy
Schwefel 4 2 0 . 9 6 8 7 𝑛 𝑛 𝑖 = 1 ( 𝑥 𝑖 s i n ( | 𝑥 𝑖 | ) [ 5 0 0 . 0 ; 5 0 0 . 0 ] 0 Multimodal
Ackley 2 0 + 𝑒 2 0 𝑒 0 . 2 ( ( 1 / 𝑛 ) 𝑛 𝑖 = 1 𝑥 2 𝑖 ) 1 / 2 𝑒 ( 1 / 𝑛 ) 𝑛 𝑖 = 1 c o s ( 2 𝜋 𝑥 𝑖 ) [ 3 0 . 0 ; 3 0 . 0 ] 0 Multimodal
Michalewicz 𝑛 𝑖 = 1 s i n ( 𝑥 𝑖 ) s i n 2 𝑚 ( ( 𝑖 𝑥 2 𝑖 ) / 𝜋 ) [ 𝜋 ; 𝜋 ] Multimodal
Himmelblau ( 𝑥 2 + 𝑥 2 1 1 1 ) 2 + ( 𝑥 1 + 𝑥 2 2 7 ) 2 + 𝑥 1 [ 5 . 0 ; 5 . 0 ] 3 . 7 8 3 9 6 Multimodal
Shubert 5 𝑖 = 1 𝑖 c o s ( ( 𝑖 + 1 ) 𝑥 1 + 𝑖 ) 5 𝑖 = 1 𝑖 c o s ( ( 𝑖 + 1 ) 𝑥 2 + 𝑖 ) [ 1 0 . 0 ; 1 0 0 ] 1 8 6 . 7 3 0 9 Multimodal