无约束 benchmark 函数

随手记录下 12 个无约束 benchmark 函数

                   

（1）Spherical函数

                                              $f_1=\sum _{i=1}^n{x}_i^2,x_i\in [-100,100]$

              全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$

                   

（2）Rosenbrock函数

                                      $f_2=\sum _{i=1}^n\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right),x_i\in [-30,30]$

              全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$。

                   

（3）Rastrigin 函数

                                      $f_3=\sum _{i=1}^n\left(x_i^2-10\cos \left(2\pi x_i\right)+10\right),x_i\in [-5.12,5.12]$

              全局最优解$x^{\ast }=(1,\dots ,1)$，$f\left(x^{\ast }\right)=0$。

                   

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（4）Griewank函数

                   $f_4=\frac{1}{4000}\sum _{i=1}^n{x}_i^2-\prod _i^n\cos \left|\frac{x_i}{\sqrt{i}}\right|+1,x_i\in [-600,600]$

              全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$。

                   

（5）Ackley 函数

                   

$f_5=-20\exp (-0.2\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2})-\exp \left(\frac{1}{n}\sum _{i=1}^n\cos \left(2\pi x_i\right)\right)+20+e,x_i\in [-32,32]$

                    

              全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$。

                   

（6）High conditioned elliptic函数

                                                         $f_6(x)=\sum _{i=1}^n{\left({10}^6\right)}^{\frac{i-1}{n-1}}{x}_i^2{x}_i\in [-100,100]$

              全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$。

                   

（7）Michalewicz函数

                                                         $f_7(x)=-\sum _{i=1}^n\sin \left(x_i\right)\sin {\left(\frac{i{x}_i^2}{\pi }\right)}^{20}{x}_i\in [0,\pi ]$

              全局最优解未知。

                   

（8）Trid函数

                                      $f_8(x)=\sum _{i=1}^n{\left(x_i-1\right)}^2-\sum _{i=2}^n{x}_i{x}_{i-1}{x}_i\in [-n^2,n^2]$

              全局最优解$i(n+1-i)$，$-\frac{n(n+4)(n-1)}{6}$。

                   

（9）Schwefel函数

                                                         $f_9=\sum _{i=1}^n\left[-x_i\sin (\sqrt{\left|x_i\right|})\right]x_i\in [-500,500]$

              全局最优解$x^{\ast }=(\text{420}\text{.9687},\dots ,\text{420}\text{.9687})$，$f\left(x^{\ast }\right)=\text{.418}\text{.9829n}$。

                   

（10）Schwefel 1.2函数

                                                         $f_{10}=\sum _{i=1}^n{\left(\sum _{j=1}^i{x}_j\right)}^2{x}_i\in [-100,100]$

              全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$。

                   

（11）Schwefel 2.4函数

                                                         $f_{11}=\sum _{i=1}^n\left[{\left(x_i-1\right)}^2+{\left(x_1-x_i^2\right)}^2\right]x_i\in [0,10]$

              全局最优解$x^{\ast }=(1,\dots ,1)$，$f\left(x^{\ast }\right)=0$。

                   

（12）Weierstrass函数

                   

$f_{12}=\sum _{i=1}^n\sum _{k=0}^{k_{\max }}\left[a^k\cos \left(2\pi b^k\left(x_i+0.5\right)\right)\right]-n\sum _{k=0}^{k_{\max }}{a}^k\cos \left(\pi b^k{x}_i\right)x_i\in [-0.5,0.5]$

                   

              其中。$a=0.5,b=3$，$k_{\max }=20$；全局最优解$x^{\ast }=(0,\dots ,0)$，$f\left(x^{\ast }\right)=0$。