Applied Computational Intelligence and Soft Computing

Research Article

Hybrid PSO-SA Type Algorithms for Multimodal Function Optimization and Reducing Energy Consumption in Embedded Systems

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