Quantum Behaved Particle Swarm Optimization with Neighborhood Search for Numerical Optimization

Table 1

Benchmark problems used in the experiments.

Problems	D	Search range
𝑓1(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖	30	[−100, 100]
𝑓2(𝑥)=∑^𝐷−1𝑖=1100𝑥^2𝑖−𝑥𝑖+1^2+𝑥𝑖−1^2	30	[−2.048, 2.048]
𝑓3(𝑥)=−20exp−0.21/𝐷∑^𝐷𝑖=1𝑥^2𝑖−exp1/𝐷∑^𝐷𝑖=1cos2𝜋𝑥𝑖+20+𝑒	30	[−32.768, 32.768]
𝑓4(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖/4000−∏^𝐷𝑖=1cos𝑥𝑖/√𝑖+1	30	[−600, 600]
𝑓5(𝑥)=∑^𝐷𝑖=1∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑥𝑖+0.5−𝐷∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘⋅0.5, 𝑎=0.5, 𝑏=3, 𝑘max=20	30	[−0.5, 0.5]
𝑓6(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	30	[−5.12, 5.12]
𝑓7(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−10cos2𝜋𝑦𝑖+10; 𝑦𝑖=𝑥𝑖,𝑥𝑖<1/2round(2𝑥𝑖)/2,𝑥𝑖≥1/2	30	[−5.12, 5.12]
𝑓8(𝑥)=418.9829⋅𝐷−∑^𝐷𝑖=1𝑥𝑖sin𝑥𝑖	30	[−500, 500]
𝑓9(𝑥)=−20exp−0.21/𝐷∑^𝐷𝑖=1𝑦^2𝑖−exp1/𝐷∑^𝐷𝑖=1cos2𝜋𝑦𝑖+20+𝑒; 𝐲=𝐌∗𝐱	30	[−32.768, 32.768]
𝑓10(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖/4000−∏^𝐷𝑖=1cos𝑦𝑖/√𝑖+1; 𝐲=𝐌∗𝐱	30	[−600, 600]
𝑓11(𝑥)=∑^𝐷𝑖=1∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑦𝑖+0.5−𝐷∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘⋅0.5; 𝐲=𝐌∗𝐱	30	[−0.5, 0.5]
𝑓12(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−10cos2𝜋𝑦𝑖+10; 𝐲=𝐌∗𝐱	30	[−5.12, 5.12]