无约束 benchmark 函数

随手记录下 12 个无约束 benchmark 函数

                   

（1）Spherical函数

                                              \({{f}_{1}}=\sum\limits_{i=1}^{n}{x_{i}^{2}},\quad {{x}_{i}}\in [-100,100]\)

              全局最优解\(x^{*}=(0, \ldots, 0)\)，\(f\left( {{x}^{*}} \right)=0\)

                   

（2）Rosenbrock函数

                                      \({{f}_{2}}=\sum\limits_{i=1}^{n}{\left( 100{{\left( {{x}_{i+1}}-x_{i}^{2} \right)}^{2}}+{{\left( {{x}_{i}}-1 \right)}^{2}} \right)},\quad {{x}_{i}}\in [-30,30]\)

              全局最优解\(x^{*}=(0, \ldots, 0)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（3）Rastrigin 函数

                                      \({{f}_{3}}=\sum\limits_{i=1}^{n}{\left( x_{i}^{2}-10\cos \left( 2\pi {{x}_{i}} \right)+10 \right)},\quad {{x}_{i}}\in [-5.12,5.12]\)

              全局最优解\({{x}^{*}}=(1,\ldots ,1)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（4）Griewank函数

                   \({{f}_{4}}=\frac{1}{4000}\sum\limits_{i=1}^{n}{x_{i}^{2}}-\prod\limits_{i}^{n}{\cos \left| \frac{{{x}_{i}}}{\sqrt{i}} \right|}+1,\quad {{x}_{i}}\in [-600,600]\)

              全局最优解\(x^{*}=(0, \ldots, 0)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（5）Ackley 函数

                   

\({{f}_{5}}=-20\exp (-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}{x_{i}^{2}}})-\exp \left( \frac{1}{n}\sum\limits_{i=1}^{n}{\cos }\left( 2\pi {{x}_{i}} \right) \right)\text{ }+20+e,\quad {{x}_{i}}\in [-32,32]\text{ }\)

                    

              全局最优解\(x^{*}=(0, \ldots, 0)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（6）High conditioned elliptic函数

                                                         \({{f}_{6}}(x)=\sum\limits_{i=1}^{n}{{{\left( {{10}^{6}} \right)}^{\frac{i-1}{n-1}}}}x_{i}^{2}\quad \ {{x}_{i}}\in [-100,100]\text{ }\)

              全局最优解\(x^{*}=(0, \ldots, 0)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（7）Michalewicz函数

                                                         \({{f}_{7}}(x)=-\sum\limits_{i=1}^{n}{\sin }\left( {{x}_{i}} \right)\sin {{\left( \frac{ix_{i}^{2}}{\pi } \right)}^{20}}\quad {{x}_{i}}\in [0,\pi ]\text{ }\)

              全局最优解未知。

                   

（8）Trid函数

                                      \({{f}_{8}}(x)=\sum\limits_{i=1}^{n}{{{\left( {{x}_{i}}-1 \right)}^{2}}}-\sum\limits_{i=2}^{n}{{{x}_{i}}}{{x}_{i-1}}\quad {{x}_{i}}\in [-{{n}^{2}},{{n}^{2}}]\text{ }\)

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

                   

（9）Schwefel函数

                                                         \({{f}_{9}}=\sum\limits_{i=1}^{n}{\left[ -{{x}_{i}}\sin (\sqrt{\left| {{x}_{i}} \right|}) \right]}\quad {{x}_{i}}\in [-500,500]\text{ }\)

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

                   

（10）Schwefel 1.2函数

                                                         \({{f}_{10}}=\sum\limits_{i=1}^{n}{{{\left( \sum\limits_{j=1}^{i}{{{x}_{j}}} \right)}^{2}}}\quad \ {{x}_{i}}\in [-100,100]\text{ }\)

              全局最优解\(x^{*}=(0, \ldots, 0)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（11）Schwefel 2.4函数

                                                         \({{f}_{11}}=\sum\limits_{i=1}^{n}{\left[ {{\left( {{x}_{i}}-1 \right)}^{2}}+{{\left( {{x}_{1}}-x_{i}^{2} \right)}^{2}} \right]}\quad \ {{x}_{i}}\in [0,10]\text{ }\)

              全局最优解\({{x}^{*}}=(1,\ldots ,1)\)，\(f\left( {{x}^{*}} \right)=0\)。

                   

（12）Weierstrass函数

                   

\({{f}_{12}}=\sum\limits_{i=1}^{n}{\sum\limits_{k=0}^{{{k}_{\max }}}{\left[ {{a}^{k}}\cos \left( 2\pi {{b}^{k}}\left( {{x}_{i}}+0.5 \right) \right) \right]}}\text{ }-n\sum\limits_{k=0}^{{{k}_{\max }}}{{{a}^{k}}}\cos \left( \pi {{b}^{k}}{{x}_{i}} \right)\text{ }\quad \ {{x}_{i}}\in [-0.5,0.5]\text{ }\)

                   

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