An Improved Real-Coded Population-Based Extremal Optimization Method for Continuous Unconstrained Optimization Problems

Table 1

Benchmark test functions [22, 24].

Test functions	Search space	𝑁	Global optimum
𝑓1=−20exp−0.021𝑁^𝑁𝑖∑=1𝑥^2𝑖1−exp𝑁^𝑁𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒	[−30,30]^𝑁	30	0 (min)
𝑓2=0.1∑^𝑁𝑖=1cos5𝜋𝑥𝑖−∑^𝑁𝑖=1𝑥^2𝑖	[−1,1]^𝑁	30	0.1𝑁 (max)
𝑓3=exp−0.5∑^𝑁𝑖=1𝑥^2𝑖	[−1,1]^𝑁	30	1 (max)
𝑓41=1+4000^𝑁𝑖∑=1𝑥^2𝑖−∏^𝑁𝑖=1𝑥cos^𝑖√𝑖	[−600,600]^𝑁	30	0 (min)
𝑓5=10𝑁+∑^𝑁𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖	[−5.12,5.12]^𝑁	30	0 (min)
𝑓6=∑^𝑁−1𝑖=1100𝑥𝑖+1−𝑥^2𝑖^2−𝑥𝑖−1^2	[−30,30]^𝑁	30	0 (min)
𝑓7=∑^𝑁𝑖=1𝑥^2𝑖+∑^𝑁𝑖=1𝑖2𝑥𝑖^2+∑^𝑁𝑖=1𝑖2𝑥𝑖^4	[−5.12,5.12]^𝑁	30	0 (min)
𝑓8=∑^𝑁𝑖=1𝑥^2𝑖	[−5.12,5.12]^𝑁	30	0 (min)
𝑓9=∑^𝑁𝑖=1𝑖𝑥^2𝑖	[−5.12,5.12]^𝑁	30	0 (min)
𝑓10=∑^𝑁𝑖=1𝑥𝑖+∏^𝑁𝑖=1𝑥𝑖	[−10,10]^𝑁	30	0 (min)
𝑓11=−∑^𝑁𝑖=1sin𝑥𝑖sin^2𝑚𝑖𝑥^2𝑖𝜋	[0,𝜋]^𝑁	10	−9.66 (min)
𝑓12=−∑^𝑁𝑖=1𝑥𝑖sin√𝑥𝑖	[−500,500]^𝑁	30	−12569.487 (min)