Smallest-Small-World Cellular Harmony Search for Optimization of Unconstrained Benchmark Problems

Table 2

Definitions and specifications of two design variables problems.

Functions	Definition
Rosenbrock’s valley	Minimize  𝑓𝑥1,𝑥2=100(𝑥^21−𝑥^22)^2+(1−𝑥1)^2; −2.048≤𝑥1≤2.048,−2.048≤𝑥2≤2.048; 𝑥1=1,𝑥2=1,𝑓(1,1)=0
Branin’s function	Minimize  𝑓𝑥1,𝑥2=𝑎(𝑥2−𝑏𝑥^21+𝑐𝑥1−𝑑)^2+𝑒1−𝑓cos𝑥1+10; 𝑎=1,𝑏=5.1/4𝜋^2,𝑐=5/𝜋,𝑑=6,𝑒=10,𝑓=1/8𝜋; −5≤𝑥1≤10,0≤𝑥2≤15; 𝑥1,𝑥2=(−𝜋,12.275),(𝜋,2.275),(9.42478,2.475); 𝑓𝑥1,𝑥2=0.397887
Easom’s function	Minimize  𝑓𝑥1,𝑥2=−cos𝑥1cos𝑥2exp−𝑥1−𝜋^2−𝑥2−𝜋^2; −100≤𝑥1≤100,−100≤𝑥2≤100; 𝑥1,𝑥2=(𝜋,𝜋),𝑓𝑥1,𝑥2=−1
Goldstein price’s function	Minimize  𝑓𝑥1,𝑥2=[1+𝑥1+𝑥2+1)^219−14𝑥1+3𝑥^21−14𝑥2+6𝑥1𝑥2+3𝑥^22; ×30+2𝑥1−3𝑥2^218−32𝑥1+12𝑥^21+48𝑥2−36𝑥1𝑥2+27𝑥^22; −2≤𝑥1≤2,−2≤𝑥2≤2; 𝑥1,𝑥2=(0,−1),𝑓𝑥1,𝑥2=3
Six-hump camel back function	Minimize𝑓𝑥1,𝑥2=4−2.1𝑥^21+𝑥^41/3𝑥^21+𝑥1𝑥2+−4+4𝑥^22𝑥^22; −3≤𝑥1≤3,−2≤𝑥2≤2; 𝑥1,𝑥2=(−0.0898,0.7126),(0.0898,−0.7126)𝑓𝑥1,𝑥2=−1.0316