Summary

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  • Ackley Objective

    $$ f(x) = -20 \exp { -0.2\sqrt { \dfrac{1}{D} \sum_{i=1}^{D}x_i^2 } } - \exp { \dfrac{1}{D} \sum_{i=1}^{D}x_i^2 cos(2\pi x_i) } $$
    $$ x,y \in [-32,32] $$
    $$ x = 0 $$ $$ y = 0 $$
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  • Alpine Objective

    $$ f(x) = \sum_{i=1}^{D} | x_i sin(x_i) + 0.1x_i | $$
    $$ x,y \in [-10,10] $$
    $$ x = 0 $$ $$ y = 0 $$
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  • Beale Objective

    $$ f(x_1, x_2) = ( 1.5 - x_1 + x_1 x_2)^2 + ( 2.25 - x_1 + x_1 x_2^2)^2 + $$ $$ ( 2.625 - x_1 + x_1 x_2^3)^2 $$
    $$ x,y \in [-4.5,4.5] $$
    $$ x = 3 $$ $$ y = 0.5 $$
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  • Bohachevsky Objective

    $$ f(x) = \sum_{i=1}^{D} (x_i^2 + 2x_i+1^2 - 0.3cos(3\pi x_i) - 0.4cos(4\pi x_i+1) + 0.7) $$
    $$ x,y \in [-15,15] $$
    $$ x = 0 $$ $$ y = 0 $$
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  • Egg Holder Objective

    $$ f(x_1, x_2) = -(x_2 + 47)sin(\sqrt{|x_2 + \dfrac{x_1}{2} + 47|}) + $$ $$ sin(\sqrt{|x_1 - (x_2) + 47|}) (-x_1) $$
    $$ x,y \in [-512,512] $$
    $$ x = 512 $$ $$ y = 404.23181 $$
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  • Goldstein-Price Objective

    $$ f(x_1, x_2) = [1 + (x_1 + x_2 + 1)^2 x $$ $$ (19 - 14x_1 + 3x_1^2 - 14x_2 + 6x_1x_2 + 3x_2^2)] x $$ $$ [30 + (2x_1 − 3x_2)^2(18 − 32x_1 + 12x_1^2 + 48x_2 −36x_1x_2 + 27x_2^2)] $$
    $$ x,y \in [-2,2] $$
    $$ x = 0 $$ $$ y = -1 $$
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  • Griewank

    $$ f_{grw}(x)=\frac{1}{4000} \sum_{i=1}^{D} {x_{i}^{2}} - \prod_{i=1}^{D}\cos\left (\frac{x_{i}}{\sqrt{i}} \right ) +1$$
    $$ x_{i} \in [-600,600] $$
    $$ f_{grw}^{*} =f_{grw}(0,...,0)=0 $$
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  • Levy 13

    $$ f_{lvy}(x)= \sum_{i=1}^{D-1}($$ $$ \sin^{^{2}}(3\pi x_{i}) +$$ $$ (x_{i}-1)^{2}\left[1+\sin^{2}(3\pi x_{i+1}))\right] +$$ $$ (x_{i+1}-1)^{2}$$ $$ \left[1+\sin ^{^{2}}(2\pi^{x_{i+1}}))\right])$$
    $$ x_{i}\in [-10,10] \\ D >= 2 $$
    $$ f_{lvy}^{*} =f_{lvy}(1,...,1)=0 $$
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  • Michalewicz

    $$ f_{mic}(x)= -\sum_{i=1}^{D}\sin(x_{i})(\sin(ix_{i}^{2}/\pi))^{2p}$$
    $$ x_{i} \in [0,\pi] $$ $$ \text{for } p= 10$$
    $$f_{mic}^{*}\approx -1.8013 ( D=2)$$ $$f_{mic}^{*}\approx -4.6877 ( D=5)$$ $$f_{mic}^{*}\approx -96602 ( D=10)$$ $$f_{mic}^{*}\approx -29.6309 ( D=30)$$
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  • Pathological

    $$ f_{pth}(x)=\sum_{i=1}^{D}\left (0.5+\frac{\sin ^{2}\sqrt{(100x_{i}^{2}+x_{i+1}^{2})}-0.5}{1+0.001(x_{i}^{2}-2x_{i+1}x_{i+1}+x_{i+1}^{2})^{2}}\right )$$
    $$ D >= 2 $$ $$ x_{i} \in [-100,100]$$
    $$f_{pth}^{*} =f_{pth}(0,...,0)=0 $$
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  • Quadric(Schwefel 1.2)

    $$ f_{qdr}(x)=\sum_{i=1}^{D}\left (\sum_{j=i}^{i}x_{j} \right )^{2} $$
    $$ x_{i} \in [-100,100] $$
    $$f_{qdr}^{*} =f_{qdr}(0,...,0)=0 $$
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  • Quartic

    $$ f_{qrt}(x)=\sum_{i=1}^{D} i x_{i}^{4}$$
    $$ x_{i} \in [-1.28,1.28] $$
    $$f_{qrt}^{*} =f_{qrt}(0,...,0)=0 $$
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  • Rana Objective

    $$ f(x) = \sum_{i=1}^{D} x_i * sin(α)cos(Β) + (x_{i+1modD} + 1)cos(α)sin(Β) $$, where D >= 2, $$ α = \sqrt{|x_{i+1}+1-x_i} $$ and $$ Β = \sqrt{|x_{i+1}+1+x_i} $$
    $$ x \in [-512,512] $$
    $$ x = -512 $$
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  • Rastgrin Objective

    $$ f(x) = \sum_{i=1}^{D} (x_i^2 - 10cos(2\pi{}x_i) + 10) $$
    $$ x \in [-5.12,5.12] $$
    $$ x = 0 $$
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  • Rosenbrock Objective

    $$ f(x) = \sum_{i=1}^{D-1} (100(x_{i+1} - x_i^2)^2 + (x_i - 1)^2 $$, D >= 2
    $$ x \in [-2.048,2.048] $$
    $$ x = 1 $$
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  • Salomon Objective

    $$ f(x) = -cos(2\pi{}\sum_{i=1}^{D}x_i^2) + 0.1\sqrt{\sum_{i=1}^{D}x_i^2} +1 $$
    $$ x \in [-100,100] $$
    $$ x = 0 $$
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  • Schweffel 2.22 Objective

    $$ f(x) = \sum_{i=1}^{D}|x_i| + \prod_{i=1}^{D}|x_i| $$
    $$ x \in [-10,10] $$
    $$ x = 0 $$
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  • Schweffel 2.26 Objective

    $$ f(x) = -\sum_{i=1}^{D}(x_isin(\sqrt{|x_i|})) $$
    $$ x \in [-500,500] $$
    $$ x = 420.9687 $$
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  • Six-Hump Camel-Back

    $$f_{6h}(x_{1},x_{2})=4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+4x_{2}^{4} $$
    $$ x_{i} \in [-5,5] $$
    $$ f_{6h}^{*}(0.08983,-0.7126) =f_{6h}^{*}(-0.08983,-0.7126)= -1.0.316285 $$
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  • Skew Rastrigin

    $$ f_{skr}(x)=10D+\sum_{i=1}^{D}(y_{i}^{2}-10\cos(2\pi y_{i}))$$ $$ \text{ where } y_{i}=\begin{cases} 10x_{i},&\text{if}\ x_{i}=1\\ x_{i},&\text{otherwise} \end{cases} $$
    $$ x_{i} \in [-5,5] $$
    $$f_{skr}^{*} =f_{skr}(0,...,0)=0 $$
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  • Spherical

    $$ f_{sph}(x)=\sum_{i=1}^{D}x_{i}^{2} $$
    $$ x_{i} \in [-100,100] $$
    $$f_{shp}^{*} =f_{shp}(0,...,0)=0 $$
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  • Step

    $$ f_{stp}(x)=\sum_{i=1}^{D}(\left \lfloor x_{i} +0.5 \right \rfloor)^{2}$$
    $$ x_{i} \in [-20,20] $$
    $$f_{stp}^{*} =f_{stp}(0,...,0)=0 $$
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  • Weierstrass

    $$f_{wei}(x)=\sum_{i=1}^{D}\sum_{k=0}^{20}\left [ 0.5^{k}\cos(2 \pi3^{k}(x_{i}+0.5)) \right ]-$$ $$D\sum_{k=0}^{20}\left [0.5^{k}\cos(2 \pi3^{k}0.5) \right ] $$
    $$ x_{i} \in [-0.5,0.5] $$
    $$f_{wei}^{*} =f_{wei}(0,...,0)=0 $$
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  • Zakharov

    $$ f_{zak}(x)=\sum_{i=1}^{D}x_{i}^{2}+\left [ \sum_{i=1}^{D}ix_{i}/2 \right ]^{2}+\left [ \sum_{i=1}^{D}ix_{i}/2 \right ]^{4}$$
    $$ x_{i} \in [-5,10] $$ $$ D >= 2$$
    $$f_{zak}^{*} =f_{zak}(0,...,0)=0 $$
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SwarmVis - DragonBrain