Solving Unconstrained Global Optimization Problems via Hybrid Swarm Intelligence Approaches

Table 2

Numerical results obtained from the proposed RGA-PSO algorithm for solving 14 UGO problems.

TP number	Function name	Global minimum	Required accuracy	Success rate (%)	𝑓(𝐱^∗RGA-PSO)	𝑓(𝐱^meanRGA-PSO)	𝑓(𝐱^worstRGA-PSO)	ME	MCCT (sec.)
1	SHCB	−1.0316	1𝐸−4	100	−1.0316	−1.0316	−1.0316	0.00000000	40.56
2	GP	3	1𝐸−3	100	3	3	3	0.00000000	38.29
3	ES	−1	1𝐸−3	100	−1	−1	−1	0.00000000	41.29
4	B2	0	1𝐸−2	100	0.00000000	0.00000000	0.00000000	0.00000000	40.26
5	DJ	0	1𝐸−4	100	0.00000000	5.533𝐸−226	5.533𝐸−224	5.533𝐸−226	40.50
6	Booth	0	1𝐸−4	100	0.00000000	0.00000000	0.00000000	0.00000000	37.57
7	RC	5/4𝜋	1𝐸−3	100	0.39788735	0.39788735	0.39788735	0.00000000	40.97
8	RA	−2	1𝐸−4	100	−2	−2	−2	0.00000000	39.31
9	RS2	0	1𝐸−3	100	0.00000000	8.709𝐸−28	8.686𝐸−26	8.709𝐸−28	37.47
10	RS5	0	1𝐸−3	100	9.587𝐸−12	9.452𝐸−06	9.058𝐸−05	9.452𝐸−06	306.15
11	SH	−186.7309	1𝐸−2	100	−186.7309	−186.7309	−186.7309	0.00000000	54.11
12	ZA2	0	1𝐸−4	100	0.00000000	0.00000000	0.00000000	0.00000000	40.74
13	ZA5	0	1𝐸−4	100	0.00000000	2.660𝐸−117	2.660𝐸−115	2.660𝐸−117	157.09
14	ZA10	0	1𝐸−3	100	2.453𝐸−120	4.375𝐸−26	2.933𝐸−24	4.375𝐸−26	194.73

𝑓(𝐱^∗_RGA-PSO): the objective value obtained using the best RGA-PSO solution.
𝑓(𝐱^mean_RGA-PSO): mean objective value obtained using the RGA-PSO solutions.
𝑓(𝐱^worst_RGA-PSO): the worst objective value obtained using the worst RGA-PSO solution.