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Table 1 12 benchmark functions

From: An image denoising method based on BP neural network optimized by improved whale optimization algorithm

Function Dim Range \(f_{min}\)
\(F_1(x)=\sum _{i=1}^{n}x_i^2\) 30 [\(-100,100\)] 0
\(F_2(x)=\sum _{i=1}^{n}\mid x_i\mid +\prod _{i=1}^{n}\mid x_i\mid\) 30 [\(-10,10\)] 0
\(F_3(x)=(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^{2}(x_i-a_{ij})^6})^{-1}\) 2 [\(-65,65\)] 1
\(F_4(x)=max_i\{\mid x_i\mid ,1\le i\le n\}\) 30 [\(-100,100\)] 0
\(F_5(x)=\sum _{i=1}^{n-1}[100(x_{i+1}-x_i^2)^2+(x_i-1)^2]\) 30 [\(-30,30\)] 0
\(F_6(x)=\sum _{i=1}^{n}([x_i+0.5])^2\) 30 [\(-100,100\)] 0
\(F_7(x)=\sum _{i=1}^{n}ix_i^4+\mathrm{{random}}[0,1)\) 30 [\(-1.28,1.28\)] 0
\(F_8(x)=\sum _{i=1}^{n}[x_i^2-10\cos (2\pi x_i)+10]\) 30 [\(-5.12,5.12\)] 0
\(F_9(x)=\frac{1}{4000}\sum _{i=1}^{n}x_i^2-\prod _{i=1}^{n}\cos (\frac{x_i}{\sqrt{i}})+1\)   30 [\(-600,600\)] 0
\(F_{10}(x)=\frac{\pi }{n}\{10\sin (\pi y_1)+\sum _{i=1}^{n-1}(y_i-1)^2[1+10\sin _2(\pi y_{i+1})]+(y_n-1)^2\}+\sum _{i=1}^{n}u(x_i,10,100,4)\)    
\(y_i=1+\frac{x_i+1}{4}u(x_i,a,k,m)= {\left\{ \begin{array}{ll} k(x_i-a)^m \qquad \qquad x_i>a \\ 0 \qquad \qquad \qquad -a<x_i<a \\ k(-x_i-a)^m \qquad \quad x_i<-a \end{array}\right. }\) 30 [\(-50,50\)] 0
\(F_{11}(x)=(x_2-\frac{5.1}{4\pi ^2}x_1^2+\frac{5}{\pi }x_1-6)^2+10(1-\frac{1}{8\pi })\cos x_1+10\) 2 [\(-5,5\)] 0.398
\(F_{12}(x)=-\sum _{i=1}^{7}[(X-a_i)(X-a_i)^T+c_i]^{-1}\) 4 [0,10] -10.4028