A Novel Self-Adaptive Harmony Search Algorithm

Table 1

Six shifted rotated unimodal benchmark functions.

Name	Expression
Shifted rotated Cigar-Tablet's function	𝑓1=𝑧^21+ln8⋅𝑧^2𝑛+ln4⋅∑^𝑛−1𝑖=2𝑧^2𝑖+bias
Shifted rotated Rosenbrock's function	𝑓2=∑^𝑛−1𝑖=1100⋅𝑧𝑖+1−𝑧^2𝑖^2+𝑧𝑖−1^2+bias
Shifted rotated Schwefel's problem 1.2 with noise in fitness	𝑓3=∑^𝑛∑𝑖=1^𝑖𝑗=1𝑧𝑗^2(1+0.4∗|𝑁(0,1)|)+bias
Shifted rotated Schwefel’s problem 2.21 with noise in fitness	𝑓4=max𝑖𝑧𝑖(1+0.4∗|𝑁(0,1)|)+bias
Shifted rotated Schwefel’s problem 2.22 with noise in fitness	𝑓5=∑^𝑛𝑖=1𝑧𝑖+∏^𝑛𝑖=1𝑧𝑖(1+0.4∗|𝑁(0,1)|)+bias
Shifted rotated sphere function with noise in fitness	𝑓6=∑^𝑛𝑖=1𝑧^2𝑖(1+0.4∗|𝑁(0,1)|)+bias

^∗𝑧=(𝑥−𝑜)𝐌. 𝑥={𝑥₁,𝑥₂,…,𝑥_𝑛} is a solution in the search range of [−100, 100]. 𝑜={𝑜₁,𝑜₂,…,𝑜_𝑛} is a random vector in the range of [−100, 100]. 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, 390, −450, −310, −180, and −450, respectively. For the second benchmark function, global optimum 𝑥^∗=𝑜+1 and 𝑓^∗=bias; for other functions, global optimum 𝑥^∗=𝑜 and 𝑓^∗=bias.