A Novel Self-Adaptive Harmony Search Algorithm

Table 2

Six well-known multimodal benchmark functions.

Name	Expression
Shifted rotated Ackley's function	𝑓7=−20exp−0.2⋅1𝑛^𝑛𝑖∑=1𝑧^2𝑖1−exp𝑛^𝑛𝑖∑=1cos2𝜋𝑧𝑖+20+𝑒+bias
Shifted rotated Bohachevsky's function	𝑓8=∑^𝑛−1𝑖=1𝑧^2𝑖+2𝑧^2𝑖+1−0.3cos3𝜋𝑧𝑖−0.4cos4𝜋𝑧𝑖+1+0.7+bias
Shifted rotated Griewank's function	𝑓9=14000^𝑛𝑖∑=1𝑧^2𝑖−∏^𝑛𝑖=1𝑧cos^𝑖√𝑖+1+bias
Shifted rotated Rastrigin's function	𝑓10=∑^𝑛𝑖=1𝑧^2𝑖−10cos2𝜋𝑧𝑖+10+bias
Shifted rotated Schaffer's function	𝑓11=∑^𝑛−1𝑖=1𝑧^2𝑖+2𝑧^2𝑖+1^0.251+sin^250𝑧^2𝑖+2𝑧^2𝑖+1^0.1+bias
Shifted rotated Schwefel’s problem 2.13	𝑓12=∑^𝑛∑𝑖=1^𝑛𝑗=1𝑎𝑖𝑗sin𝛼𝑗+𝑏𝑖𝑗cos𝛼𝑗−∑^𝑛𝑗=1𝑎𝑖𝑗sin𝑧𝑗+𝑏𝑖𝑗cos𝑧𝑗+bias

^∗𝑧=(𝑥−𝑜)𝐌. 𝑥={𝑥₁,𝑥₂,…,𝑥_𝑛} is a solution in the search range of [−10, 10]. 𝑜={𝑜₁,𝑜₂,…,𝑜_𝑛} is a random vector in the range of [−10, 10]. M is an identity orthogonal matrix. bias∈ℝ is also a random number. In this study, these biases are randomly given. They are equal to −140, −310, −180, −330, 390, and −460, respectively. For the twelfth benchmark function, global optimum 𝑥^∗=𝛼 and 𝑓^∗=bias; for other functions, global optimum 𝑥^∗=𝑜 and 𝑓^∗=bias.