Visualizations of problem landscapes

Plots

Show plots in columns (click on Dimension/Function/Instance/Visualization type below to show all plots for the chosen category)

Dimension	Function	Instance	Visualization type

Plot explanation

Search space cuts

The plots with search space cuts show the function value f along various lines in the search space that go through the global optimum \mathbf{x}_\mathrm{opt}. The colored lines change the value of only one variable x_i at a time keeping the rest fixed to \mathbf{x}_\mathrm{opt}. The gray line represents the line that goes through \mathbf{x}_\mathrm{opt} in the direction of the all-ones vector (i.e., in the diagonal direction). To improve visibility, only five colored lines are shown in the larger dimensions (D \geq 10), corresponding to x_1, x_2, x_{\lfloor D/2 \rfloor}, x_{D-1} and x_D, where D is the search space dimension. The plots are shown in three variants:

lin-lin: both axes are linear,
lin-log: the x-axis is linear, the y-axis shows the difference between f and the optimal value f_\mathrm{opt} on a logarithmic scale,
log-log: both axes are logarithmic, the x-axis shows the absolute difference to \mathbf{x}_\mathrm{opt} (positive directions presented as x_i and negative as -x_i), the y-axis shows the difference between f and f_\mathrm{opt}.

Function value heatmap

The function value heatmap shows the function values f on a 2-D view of the search space that contains the optimal solution and is approximated by a grid. In addition to color-coded function values, the plots include level sets in gray hues. For dimensions larger than 2, the heatmaps of pairs of variables are organized into a matrix. To improve visibility, only five variables are included in the matrix in the larger dimensions (D \geq 10), corresponding to x_1, x_2, x_{\lfloor D/2 \rfloor}, x_{D-1} and x_D, where D is the search space dimension.

Normalized rank heatmap

The normalized rank heatmap shows, instead of absolute function values f, their normalized rank with 0 corresponding to the best rank and 1 to the worst one on a 2-D view of the search space that contains the optimal solution and is approximated by a grid. In addition to color-coded ranks, the plots include level sets in gray hues. For dimensions larger than 2, the heatmaps of pairs of variables are organized into a matrix. To improve visibility, only five variables are included in the matrix in the larger dimensions (D \geq 10), corresponding to x_1, x_2, x_{\lfloor D/2 \rfloor}, x_{D-1} and x_D, where D is the search space dimension.

Surface plot

The surface plot shows the function values f on a 3-D view of the search space and is available only for 2-D problems. To improve visibility, the z-axis is inverted, so that the global optimum is at the top of the plot.

Problem definition

Functions with moderate noise

Sphere

f_\mathrm{sphere}(\mathbf{x}) = \|\mathbf{z}\|^2

\mathbf{z}= \mathbf{x}- \mathbf{x^\mathrm{opt}}

Properties:

Presumably the most easy continuous domain search problem, given the volume of the searched solution is small (i.e. where pure monte-carlo random search is too expensive).

unimodal
highly symmetric, in particular rotationally invariant

f₁₀₁: Sphere with moderate gaussian noise

f_{101}(\mathbf{x}) = f_{\mathrm{GN}}({f_\mathrm{sphere}(\mathbf{x}),0.01}) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₀₂: Sphere with moderate uniform noise

f_{102}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{sphere}(\mathbf{x}),0.01\left(0.49 + \dfrac{1}{D}\right),0.01}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₀₃: Sphere with moderate seldom cauchy noise

f_{103}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{sphere}(\mathbf{x}),0.01,0.05}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Rosenbrock

f_\mathrm{rosenbrock}(\mathbf{x}) = \sum_{i = 1}^{D-1} 100\,\left(z_i^2 - z_{i+1}\right)^2 + (z_i-1)^2

\mathbf{z}= \max\!\left(1,\frac{\sqrt{D}}{8}\right)(\mathbf{x}- \mathbf{x^\mathrm{opt}}) + 1
\mathbf{z^\mathrm{opt}}=\mathbf{1}

Properties:

So-called banana function due to its 2-D contour lines as a bent ridge (or valley). In the beginning, the prominent first term of the function definition attracts to the point \mathbf{z}=\mathbf{0}. Then, a long bending valley needs to be followed to reach the global optimum. The ridge changes its orientation D-1 times.

in larger dimensions the function has a local optimum with an attraction volume of about 25%

f₁₀₄: Rosenbrock with moderate gaussian noise

f_{104}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{rosenbrock}(\mathbf{x}),0.01}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₀₅: Rosenbrock with moderate uniform noise

f_{105}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{rosenbrock}(\mathbf{x}),0.01\left(0.49 + \dfrac{1}{D}\right),0.01}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₀₆: Rosenbrock with moderate seldom cauchy noise

f_{106}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{rosenbrock}(\mathbf{x}),0.01,0.05}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Functions with severe noise

Sphere

f_\mathrm{sphere}(\mathbf{x}) = \|\mathbf{z}\|^2

\mathbf{z}= \mathbf{x}- \mathbf{x^\mathrm{opt}}

Properties:

Presumably the most easy continuous domain search problem, given the volume of the searched solution is small (i.e. where pure monte-carlo random search is too expensive).

unimodal
highly symmetric, in particular rotationally invariant

f₁₀₇: Sphere with gaussian noise

f_{107}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{sphere}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₀₈: Sphere with uniform noise

f_{108}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{sphere}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₀₉: Sphere with seldom cauchy noise

f_{109}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{sphere}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Rosenbrock

f_\mathrm{rosenbrock}(\mathbf{x}) = \sum_{i = 1}^{D-1} 100\,\left(z_i^2 - z_{i+1}\right)^2 + (z_i-1)^2

\mathbf{z}= \max\!\left(1,\frac{\sqrt{D}}{8}\right)(\mathbf{x}- \mathbf{x^\mathrm{opt}}) + 1
\mathbf{z^\mathrm{opt}}=\mathbf{1}

Properties:

a local optimum with an attraction volume of about 25%

f₁₁₀: Rosenbrock with gaussian noise

f_{110}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{rosenbrock}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₁₁: Rosenbrock with uniform noise

f_{111}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{rosenbrock}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₁₂: Rosenbrock with seldom cauchy noise

f_{112}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{rosenbrock}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Step ellipsoid

f_\mathrm{step}(\mathbf{x}) = 0.1 \max\left(|\hat{z}_1|/10^4,\, \sum_{i = 1}^{D} 10^{2\frac{i-1}{D-1}} z_i^2\right)

\hat{\mathbf{z}} = \Lambda^{\!10}\mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}})
\tilde{z}_i = \begin{cases} \lfloor0.5+\hat{z}_i\rfloor & \text{if~} \hat{z}_i > 0.5 \\ {\lfloor0.5+10\,\hat{z}_i\rfloor}/{10} & \text{otherwise} \end{cases} for i=1,\dots,D,
denotes the rounding procedure in order to produce the plateaus.
\mathbf{z}= \mathbf{Q}\tilde{\mathbf{z}}

Properties:

The function consists of many plateaus of different sizes. Apart from a small area close to the global optimum, the gradient is zero almost everywhere.

condition number is about 100

f₁₁₃: Step ellipsoid with gaussian noise

f_{113}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{step}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₁₄: Step ellipsoid with uniform noise

f_{114}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{step}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₁₅: Step ellipsoid with seldom cauchy noise

f_{115}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{step}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Ellipsoid

f_\mathrm{ellipsoid}(\mathbf{x}) = \sum_{i = 1}^{D} 10^{4\frac{i-1}{D-1}}z_i^2

\mathbf{z}= T_\mathrm{\hspace*{-0.01emosz}}(\mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}}))

Properties:

Globally quadratic ill-conditioned function with smooth local irregularities.

condition number is 10^4

f₁₁₆: Ellipsoid with gaussian noise

f_{116}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{ellipsoid}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₁₇: Ellipsoid with uniform noise

f_{117}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{ellipsoid}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₁₈: Ellipsoid with seldom cauchy noise

f_{118}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{ellipsoid}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Different Powers

f_\mathrm{diffpowers}(\mathbf{x}) = \sqrt{\sum_{i = 1}^{D}|z_i|^{2+4\frac{i - 1}{D- 1}}}

\mathbf{z}= \mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}})

Properties:

Due to the different exponents the sensitivies of the z_i-variables become more and more different when approaching the optimum.

f₁₁₉: Different Powers with gaussian noise

f_{119}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{diffpowers}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₀: Different Powers with uniform noise

f_{120}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{diffpowers}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₁: Different Powers with seldom cauchy noise

f_{121}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{diffpowers}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₂: Schaffer’s F7 with gaussian noise

f_{122}(\mathbf{x}) = f_{\mathrm{GN}\left({f_\mathrm{schaffer}(\mathbf{x}),1}\right)} + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₃: Schaffer’s F7 with uniform noise

f_{123}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{schaffer}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₄: Schaffer’s F7 with seldom cauchy noise

f_{124}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{schaffer}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Composite Griewank-Rosenbrock

f_\mathrm{f8f2}(\mathbf{x}) = \frac{1}{D-1} \sum_{i=1}^{D-1} \left(\frac{s_i}{4000} - \cos(s_i)\right) + 1

\mathbf{z}= \max\!\left(1,\frac{\sqrt{D}}{8}\right)\mathbf{R}\mathbf{x}+ 0.5
s_i = 100\,(z_i^2 - z_{i+1})^2 + (z_i-1)^2 for i=1,\dots,D
\mathbf{z^\mathrm{opt}}=\mathbf{1}

Properties:

Resembling the Rosenbrock function in a highly multimodal way.

f₁₂₅: Composite Griewank-Rosenbrock with gaussian noise

f_{125}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{f8f2}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₆: Composite Griewank-Rosenbrock with uniform noise

f_{126}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{f8f2}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₇: Composite Griewank-Rosenbrock with seldom cauchy noise

f_{127}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{f8f2}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Gallagher’s Gaussian Peaks, globally rotated

f_\mathrm{gallagher}(\mathbf{x}) = T_\mathrm{\hspace*{-0.01emosz}}\left(10 - \max_{i=1}^{101} w_i \exp\left(-\frac{1}{2D}\, (\mathbf{x}-\mathbf{y}_i)^{\mathrm{T}}\mathbf{R}^{\mathrm{T}} \mathbf{C}_i \mathbf{R}(\mathbf{x}-\mathbf{y}_i) \right)\right)^2

w_i = \begin{cases} 1.1 + 8 \times\dfrac{i-2}{99} & \text{for~} i=2,\dots,101 \\ 10 & \text{for~} i = 1 \end{cases}, three optima have a value larger than 9
\mathbf{C}_i=\Lambda^{\!\alpha_i}/\alpha_i^{1/4} where \Lambda^{\!\alpha_i} is defined as usual, but with randomly permuted diagonal elements. For i=2,\dots,101, \alpha_i is drawn uniformly randomly from the set \left\{1000^{2\frac{j}{99}} ~|~ j = 0,\dots,99\right\} without replacement, and \alpha_i=1000 for i=1.
the local optima \mathbf{y}_i are uniformly drawn from the domain [-4.9,4.9]^D for i=2,\dots,101 and \mathbf{y}_{1}\in[-4,4]^D. The global optimum is at \mathbf{x^\mathrm{opt}}=\mathbf{y}_1.

Properties:

The function consists of 101 optima with position and height being unrelated and randomly chosen.

condition number around the global optimum is about 30
same overall rotation matrix

f₁₂₈: Gallagher’s Gaussian Peaks 101-me with gaussian noise

f_{128}(\mathbf{x}) = f_{\mathrm{GN}}\left({f_\mathrm{gallagher}(\mathbf{x}),1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₂₉: Gallagher’s Gaussian Peaks 101-me with uniform noise

f_{129}(\mathbf{x}) = f_{\mathrm{UN}}\left({f_\mathrm{gallagher}(\mathbf{x}),0.49 + \dfrac{1}{D},1}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

f₁₃₀: Gallagher’s Gaussian Peaks 101-me with seldom cauchy noise

f_{130}(\mathbf{x}) = f_{\mathrm{CN}}\left({f_\mathrm{gallagher}(\mathbf{x}),1,0.2}\right) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

Plots

Plot explanation

Search space cuts

Function value heatmap

Normalized rank heatmap

Surface plot

Problem definition

Functions with moderate noise

Sphere

f101: Sphere with moderate gaussian noise

f102: Sphere with moderate uniform noise

f103: Sphere with moderate seldom cauchy noise

Rosenbrock

f104: Rosenbrock with moderate gaussian noise

f105: Rosenbrock with moderate uniform noise

f106: Rosenbrock with moderate seldom cauchy noise

Functions with severe noise

Sphere

f107: Sphere with gaussian noise

f108: Sphere with uniform noise

f109: Sphere with seldom cauchy noise

Rosenbrock

f110: Rosenbrock with gaussian noise

f111: Rosenbrock with uniform noise

f112: Rosenbrock with seldom cauchy noise

Step ellipsoid

f113: Step ellipsoid with gaussian noise

f114: Step ellipsoid with uniform noise

f115: Step ellipsoid with seldom cauchy noise

Ellipsoid

f116: Ellipsoid with gaussian noise

f117: Ellipsoid with uniform noise

f118: Ellipsoid with seldom cauchy noise

Different Powers

f119: Different Powers with gaussian noise

f120: Different Powers with uniform noise

f121: Different Powers with seldom cauchy noise

Highly multi-modal functions with severe noise

Schaffer’s F7

f122: Schaffer’s F7 with gaussian noise

f123: Schaffer’s F7 with uniform noise

f124: Schaffer’s F7 with seldom cauchy noise

Composite Griewank-Rosenbrock

f125: Composite Griewank-Rosenbrock with gaussian noise

f126: Composite Griewank-Rosenbrock with uniform noise

f127: Composite Griewank-Rosenbrock with seldom cauchy noise

Gallagher’s Gaussian Peaks, globally rotated

f128: Gallagher’s Gaussian Peaks 101-me with gaussian noise

f129: Gallagher’s Gaussian Peaks 101-me with uniform noise

f130: Gallagher’s Gaussian Peaks 101-me with seldom cauchy noise

f₁₀₁: Sphere with moderate gaussian noise

f₁₀₂: Sphere with moderate uniform noise

f₁₀₃: Sphere with moderate seldom cauchy noise

f₁₀₄: Rosenbrock with moderate gaussian noise

f₁₀₅: Rosenbrock with moderate uniform noise

f₁₀₆: Rosenbrock with moderate seldom cauchy noise

f₁₀₇: Sphere with gaussian noise

f₁₀₈: Sphere with uniform noise

f₁₀₉: Sphere with seldom cauchy noise

f₁₁₀: Rosenbrock with gaussian noise

f₁₁₁: Rosenbrock with uniform noise

f₁₁₂: Rosenbrock with seldom cauchy noise

f₁₁₃: Step ellipsoid with gaussian noise

f₁₁₄: Step ellipsoid with uniform noise

f₁₁₅: Step ellipsoid with seldom cauchy noise

f₁₁₆: Ellipsoid with gaussian noise

f₁₁₇: Ellipsoid with uniform noise

f₁₁₈: Ellipsoid with seldom cauchy noise

f₁₁₉: Different Powers with gaussian noise

f₁₂₀: Different Powers with uniform noise

f₁₂₁: Different Powers with seldom cauchy noise

f₁₂₂: Schaffer’s F7 with gaussian noise

f₁₂₃: Schaffer’s F7 with uniform noise

f₁₂₄: Schaffer’s F7 with seldom cauchy noise

f₁₂₅: Composite Griewank-Rosenbrock with gaussian noise

f₁₂₆: Composite Griewank-Rosenbrock with uniform noise

f₁₂₇: Composite Griewank-Rosenbrock with seldom cauchy noise

f₁₂₈: Gallagher’s Gaussian Peaks 101-me with gaussian noise

f₁₂₉: Gallagher’s Gaussian Peaks 101-me with uniform noise

f₁₃₀: Gallagher’s Gaussian Peaks 101-me with seldom cauchy noise