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

Separable functions

f₁: Sphere function

f_{1}(\mathbf{x}) = \|\mathbf{z}\|^2 + f_\mathrm{opt}

• \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, scale invariant

Information gained from this function:

• What is the optimal convergence rate of an algorithm?

f₂: Ellipsoidal function

f_{2}(\mathbf{x}) = \sum_{i = 1}^{D} 10^{6\frac{i-1}{D-1}}\,z_i^2 + f_\mathrm{opt}\\

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

Properties:

Globally quadratic and ill-conditioned function with smooth local irregularities.

• unimodal

• conditioning is about 10^6

Information gained from this function:

• In comparison to f₁₀: Is separability exploited?

f₃: Rastrigin function

f_{3}(\mathbf{x}) = 10 \left(D- \sum_{i = 1}^{D}\cos(2\pi z_i)\right) + \|\mathbf{z}\|^2 + f_\mathrm{opt}

• \mathbf{z}= \Lambda^{\!10}T^{{0.2}}_{\mathrm{asy}}(T_\mathrm{\hspace*{-0.01emosz}}(\mathbf{x}-\mathbf{x^\mathrm{opt}}))

Properties:

Highly multimodal function with a comparatively regular structure for the placement of the optima. The transformations T^{{}}_\mathrm{asy} and T_\mathrm{\hspace*{-0.01em}osz} alleviate the symmetry and regularity of the original Rastrigin function

• roughly 10^D local optima

• conditioning is about 10

Information gained from this function:

• In comparison to f₁₅: Is separability exploited?

f₄: Büche-Rastrigin function

f_{4}(\mathbf{x}) = 10 \left(D- \sum_{i = 1}^{D}\cos(2\pi z_i)\right) + \sum_{i = 1}^{D}z_i^2 + 100\,f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• z_i = s_i\,T_\mathrm{\hspace*{-0.01emosz}}(x_i - x_i^\mathrm{opt}) \quad \text{for} \; i = 1 \ldots D

• s_i = \begin{cases} 10\times10^{\frac{1}{2}\,\frac{i-1}{D-1}} & \text{~if~} z_i>0 \text{~and~} i = 1,3,5,\ldots \\ 10^{\frac{1}{2}\frac{i-1}{D-1}} & \text{~otherwise~} \end{cases} for i= 1,\dots,D

Properties:

Highly multimodal function with a structured but highly asymmetric placement of the optima. Constructed as a deceptive function for symmetrically distributed search operators.

• roughly 10^D local optima, conditioning is about 10, skew factor is about 10 in x-space and 100 in f-space

Information gained from this function:

• In comparison to f₃: What is the effect of asymmetry?

f₅: Linear slope

f_{5}(\mathbf{x}) = \sum_{i = 1}^{D} 5\,|s_i| - s_i z_i + f_\mathrm{opt}

• z_i = x_i if x_i^\mathrm{opt}x_i < 5^2 and z_i = x_i^\mathrm{opt} otherwise, for i=1,\dots,D. That is, if x_i exceeds x_i^\mathrm{opt} it will mapped back into the domain and the function appears to be constant in this direction.

• s_i = \mathrm{{sign}}\left(x_i^\mathrm{opt}\right)\, 10^{\frac{i-1}{D-1}} for i=1,\dots,D.

• \mathbf{x^\mathrm{opt}}= \mathbf{z^\mathrm{opt}}= 5\times\mathbf{1_-^+}

Properties:

Purely linear function testing whether the search can go outside the initial convex hull of solutions right into the domain boundary.

• \mathbf{x}^\mathrm{opt} is on the domain boundary

Information gained from this function:

• Can the search go outside the initial convex hull of solutions into the domain boundary? Can the step size be increased accordingly?

Functions with low or moderate conditioning

f₆: Attractive sector function

f_{6}(\mathbf{x}) = T_\mathrm{\hspace*{-0.01emosz}}\left(\sum_{i = 1}^{D} (s_i z_i)^2\right)^{0.9} + f_\mathrm{opt}

• \mathbf{z}= \mathbf{Q}\Lambda^{\!10}\mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}})

• s_i = \begin{cases} 10^2& \text{if~} z_i\times x_i^\mathrm{opt}> 0\\ 1 & \text{otherwise} \end{cases}

Properties:

Highly asymmetric function, where only one “hypercone” (with angular base area) with a volume of roughly {1}/{2^D} yields low function values. The optimum is located at the tip of this cone.

• unimodal

Information gained from this function:

• In comparison to f₁: What is the effect of a highly asymmetric landscape?

f₇: Step ellipsoidal function

f_{7}(\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) + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• \hat{\mathbf{z}} = \Lambda^{\!10}\mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}})

• \hat{z_i} = \begin{cases} \lfloor0.5+\hat{z}_i\rfloor & \text{if~} \left|\hat{z}_i\right| > 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.

• unimodal, non-separable, conditioning is about 100

Information gained from this function:

• Does the search get stuck on plateaus?

f₈: Rosenbrock function, original

f_{8}(\mathbf{x}) = \sum_{i = 1}^{D-1} \left( 100\,\left(z_i^2 - z_{i+1}\right)^2 + (z_i-1)^2 \right) + f_\mathrm{opt}

• \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) (Rosenbrock 1960). 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. Exceptionally, here \mathbf{x^\mathrm{opt}}\in[-3,3]^D.

• tri-band dependency structure, in larger dimensions the function has a local optimum with an attraction volume of about 25%

Information gained from this function:

• Can the search follow a long path with D-1 changes in the direction?

f₉: Rosenbrock function, rotated

f_{9}(\mathbf{x}) = \sum_{i = 1}^{D-1} \left( 100\,\left(z_i^2 - z_{i+1}\right)^2 + (z_i-1)^2 \right) + f_\mathrm{opt}

• \mathbf{z}= \max\left(1,\frac{\sqrt{D}}{8}\right)\mathbf{R}\mathbf{x}+ \mathbf{1}/2

• \mathbf{z^\mathrm{opt}}=\mathbf{1}

Properties:

rotated version of the previously defined Rosenbrock function.

Information gained from this function:

• In comparison to f₈: Can the search follow a long path with D-1 changes in the direction without exploiting partial separability?

Functions with high conditioning and unimodal

f₁₀: Ellipsoidal function

f_{10}(\mathbf{x}) = \sum_{i = 1}^{D} 10^{6\frac{i-1}{D-1}}z_i^2 + f_\mathrm{opt}

• \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, non-separable counterpart to f_2.

• unimodal, conditioning is 10^6

Information gained from this function:

• In comparison to f₂: What is the effect of rotation (non-separability)?

f₁₁: Discus function

f_{11}(\mathbf{x}) = 10^6 z_1^2 + \sum_{i = 2}^{D} z_i^2 + f_\mathrm{opt}

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

Properties:

Globally quadratic function with local irregularities. A single direction in search space is a thousand times more sensitive than all others.

• conditioning is about 10^6

Information gained from this function:

• In comparison to f₁: What is the effect of constraint-like penalization?

f₁₂: Bent cigar function

f_{12}(\mathbf{x}) = z_1^2 + 10^6\sum_{i = 2}^{D} z_i^2 + f_\mathrm{opt}

• \mathbf{z}= \mathbf{R}\,T^{{0.5}}_\mathrm{asy}(\mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}}))

Properties:

A ridge defined as \sum_{i=2}^{D} z_i^2 =0 needs to be followed. The ridge is smooth but very narrow. Due to T^{{1/2}}_\mathrm{asy} the overall shape deviates remarkably from being quadratic.

• conditioning is about 10^6, rotated, unimodal

Information gained from this function:

• Can the search continuously change its search direction?

f₁₃: Sharp ridge function

f_{13}(\mathbf{x}) = z_1^2 + 100\,\sqrt{\sum_{i = 2}^{D} z_i^2} + f_\mathrm{opt}

• \mathbf{z}= \mathbf{Q}\Lambda^{\!10}\mathbf{R}(\mathbf{x}- \mathbf{x^\mathrm{opt}})

Properties:

As for the Bent Cigar function, a ridge defined as \sum_{i=2}^D z_i^2 = 0 must be followed. The ridge is non-differentiable and the gradient is constant when the ridge is approached from any given point. Following the gradient becomes ineffective close to the ridge where the ridge needs to be followed in z_1-direction to its optimum. The necessary change in “search behavior” close to the ridge is difficult to diagnose, because the gradient towards the ridge does not flatten out.
Information gained from this function:

• In comparison to f₁₂: What is the effect of non-smoothness, non-differentiabale ridge?

f₁₄: Different powers function

f_{14}(\mathbf{x}) = \sqrt{\sum_{i = 1}^{D}|z_i|^{2+4\frac{i - 1}{D- 1}}} + f_\mathrm{opt}

• \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.

Multi-modal functions with adequate global structure

f₁₅: Rastrigin function

f_{15}(\mathbf{x}) = 10 \left(D- \sum_{i = 1}^{D}\cos(2\pi z_i)\right) + \|\mathbf{z}\|^2 + f_\mathrm{opt}

• \mathbf{z}= \mathbf{R}\Lambda^{\!10}\mathbf{Q}\,T^{{0.2}}_\mathrm{asy}(T_\mathrm{\hspace*{-0.01emosz}}(\mathbf{R}(\mathbf{x}-\mathbf{x^\mathrm{opt}})))

Properties:

Prototypical highly multimodal function which has originally a very regular and symmetric structure for the placement of the optima. The transformations T^{{}}_\mathrm{asy} and T_\mathrm{\hspace*{-0.01em}osz} alleviate the symmetry and regularity of the original Rastrigin function.

• non-separable less regular counterpart of f_3

• roughly 10^D local optima

• conditioning is about 10

Information gained from this function:

• in comparison to f₃: What is the effect of non-separability for a highly multimodal function?

f₁₆: Weierstrass function

f_{16}(\mathbf{x}) = 10 \left( \frac{1}{D} \sum_{i = 1}^{D}\sum_{k = 0}^{11} 1/2^k \cos(2\pi3^k(z_i + 1/2)) - f_0 \right)^3 + \frac{10}{D}f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• \mathbf{z}= \mathbf{R}\Lambda^{\!1/100}\mathbf{Q}\,T_\mathrm{\hspace*{-0.01emosz}}(\mathbf{R}(\mathbf{x}-\mathbf{x^\mathrm{opt}}))

• f_0 = \sum_{k = 0}^{11} 1/2^k \cos(2\pi3^k 1/2)

Properties:

Highly rugged and moderately repetitive landscape, where the global optimum is not unique.

• the term \sum_k 1/2^k \cos(2\pi3^k\dots) introduces the ruggedness, where lower frequencies have a larger weight 1/2^k.

• rotated, locally irregular, non-unique global optimum

Information gained from this function:

• in comparison to f₁₇: Does ruggedness or a repetitive landscape deter the search behavior?

f₁₇: Schaffers F7 function

f_{17}(\mathbf{x}) = \left(\frac{1}{D- 1}\sum_{i = 1}^{D- 1} \sqrt{s_i} + \sqrt{s_i} \sin^2\!\left(50\,s_i^{1/5}\right)\right)^2 + 10\,f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• \mathbf{z}= \Lambda^{\!10}\mathbf{Q}\,T^{{0.5}}_\mathrm{asy}(\mathbf{R}(\mathbf{x}-\mathbf{x^\mathrm{opt}}))

• s_i = \sqrt{z_i^2 + z_{i+1}^2} for i=1,\dots,D

Properties:

A highly multimodal function where frequency and amplitude of the modulation vary.

• asymmetric, rotated

• conditioning is low

f₁₈: Schaffers F7 function, moderately ill-conditioned

f_{18}(\mathbf{x}) = \left(\frac{1}{D- 1}\sum_{i = 1}^{D- 1} \sqrt{s_i} + \sqrt{s_i} \sin^2\!\left(50\,s_i^{1/5}\right)\right)^2 + 10\,f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• \mathbf{z}= \Lambda^{\!1000}\mathbf{Q}\,T^{{0.5}}_\mathrm{asy}(\mathbf{R}(\mathbf{x}-\mathbf{x^\mathrm{opt}}))

• s_i = \sqrt{z_i^2 + z_{i+1}^2} for i=1,\dots,D

Properties:

Moderately ill-conditioned counterpart to f_{17}

• conditioning of about 1000

Information gained from this function:

• In comparison to f₁₇: What is the effect of ill-conditioning?

f₁₉: Composite Griewank-Rosenbrock function F8F2

f_{19}(\mathbf{x}) = \frac{10}{D-1} \sum_{i=1}^{D-1} \left( \frac{s_i}{4000} - \cos(s_i) \right) + 10 + f_\mathrm{opt}

• \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.

Multi-modal functions with weak global structure

f₂₀: Schwefel function

f_{20}(\mathbf{x}) = - \frac{1}{100D}% % kept in the final print \sum_{i = 1}^{D}z_i\sin(\sqrt{|z_i|}) + 4.189828872724339 + 100f_{\mathrm{pen}}(\mathbf{z}/100) + f_\mathrm{opt}

• \hat{\mathbf{x}} = 2\times\mathbf{1_-^+}\otimes\mathbf{x}

• \hat{z}_{1} = \hat{x}_{1},\quad \hat{z}_{i+1} = \hat{x}_{i+1} + 0.25 \left({\hat{x}_{i}} - 2|x_i^{\text{opt}}| \right),\quad \text{for } i = 1, \ldots, D - 1

• \mathbf{z}= 100 (\Lambda^{10} (\hat{\mathbf{z}} - 2\left|\mathbf{x^\mathrm{opt}}\right|) + 2\left|\mathbf{x^\mathrm{opt}}\right|)

• \mathbf{x^\mathrm{opt}}= 4.2096874633/2 \;\mathbf{1_-^+}, where \mathbf{1}_-^+ is the same realization as above

Properties:

The most prominent 2^D minima are located comparatively close to the corners of the unpenalized search area (Schwefel 1981).

• the penalization is essential, as otherwise more and better minima occur further away from the search space origin

f₂₁: Gallagher’s Gaussian 101-me peaks function

f_{21}(\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 + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• 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 (see Symbols and definitions),

• 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 [-5,5]^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 (different for each instantiation of the function), based on (Gallagher and Yuan 2006).

• the conditioning around the global optimum is about 30

Information gained from this function:

• Is the search effective without any global structure?

f₂₂: Gallagher’s Gaussian 21-hi peaks function

f_{22}(\mathbf{x}) = T_\mathrm{\hspace*{-0.01emosz}}\left( 10 - \max_{i=1}^{21} 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 + f_{\mathrm{pen}}(\mathbf{x}) + f_\mathrm{opt}

• w_i = \begin{cases} 1.1 + 8 \times\dfrac{i-2}{19} & \text{for~} i=2,\dots,21 \\ 10 & \text{for~} i = 1 \end{cases}, two 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 (see Symbols and definitions),

• but with randomly permuted diagonal elements. For i=2,\dots,21, \alpha_i is drawn uniformly randomly from the set \left\{1000^{2\frac{j}{19}} ~|~ j = 0,\dots,19\right\} without replacement, and \alpha_i=1000^2 for i=1.

• the local optima \mathbf{y}_i are uniformly drawn from the domain [-4.9,4.9]^D for i=2,\dots,21 and \mathbf{y}_{1}\in [-3.92,3.92]^D. The global optimum is at \mathbf{x^\mathrm{opt}}=\mathbf{y}_1.

Properties:

The function consists of 21 optima with position and height being unrelated and randomly chosen (different for each instantiation of the function), based on (Gallagher and Yuan 2006).

• the conditioning around the global optimum is about 1000

Information gained from this function:

• In comparison to f₂₁: What is the effect of higher condition?

f₂₃: Katsuura function

f_{23}(\mathbf{x}) = \frac{10}{D^{2}} \prod_{i=1}^D\left(1 + i\sum_{j=1}^{32} \frac{\left|2^j z_i - [2^j z_i]\right|}{2^j} \right)^{10/D^{1.2}} - \frac{10}{D^{2}} + f_{\mathrm{pen}}(\mathbf{x}) + f_{\mathrm{opt}}

• \mathbf{z}= \mathbf{Q}\,\Lambda^{\!100}\mathbf{R}(\mathbf{x}-\mathbf{x^\mathrm{opt}})

Properties:

Highly rugged and highly repetitive function with more than 10^D global optima, based on the idea in (Katsuura 1991)

f₂₄: Lunacek bi-Rastrigin function

f_{24}(\mathbf{x}) = \mathrm{min}\left(\sum_{i = 1}^{D}(\hat{x}_i - \mu_0)^2, d\,D+ s\sum_{i = 1}^{D}(\hat{x}_i - \mu_1)^2\right) + 10\left(D- \sum_{i=1}^D\cos(2\pi z_i)\right) + 10^4\,f^{{}_\mathrm{pen}}(\mathbf{x}) + f_{\mathrm{opt}}

• \hat{\mathbf{x}} = 2\,\mathrm{{sign}}(\mathbf{x^\mathrm{opt}})\otimes\mathbf{x}, \mathbf{x^\mathrm{opt}}= \frac{\mu_0}{2} \mathbf{1_-^+}

• \mathbf{z}= \mathbf{Q}\Lambda^{\!100}\mathbf{R}(\hat{\mathbf{x}}-\mu_0\,\mathbf{1})

• \mu_0 = 2.5, \mu_1 = -\sqrt{\dfrac{\mu_0^2-d}{s}}, s = 1 - \dfrac{1}{2\sqrt{D+20}-8.2}, d=1

Properties:

Highly multimodal function based on (Lunacek, Whitley, and Sutton 2008) with two funnels around \frac{\mu_0}{2}\mathbf{1_-^+} and \frac{\mu_1}{2}\mathbf{1_-^+} being superimposed by the cosine. Presumably different approaches need to be used for “selecting the funnel” and for searching the highly multimodal function “within” the funnel. The function was constructed to be deceptive for evolutionary algorithms with large population size.

• the funnel of the local optimum at \frac{\mu_1}{2}\mathbf{1_-^+} has roughly 70\% of the search space volume within [-5,5]^D.

Information gained from this function: Can the search behavior be local on the global scale but global on a local scale?

References

Gallagher, Marcus, and Bo Yuan. 2006. “A General-Purpose Tunable Landscape Generator.” IEEE Transactions on Evolutionary Computation 10 (5): 590–603. https://doi.org/10.1109/TEVC.2005.863628.

Katsuura, Hidefumi. 1991. “Continuous Nowhere-Differentiable Functions – an Application of Contraction Mappings.” The American Mathematical Monthly 98 (5): 411–16. https://doi.org/10.1080/00029890.1991.12000778.

Lunacek, Monte, Darrell Whitley, and Andrew Sutton. 2008. “The Impact of Global Structure on Search.” In Proceedings of the International Conference on Parallel Problem Solving from Nature (PPSN X), 5199:498–507. Lecture Notes in Computer Science. Springer. https://doi.org/10.1007/978-3-540-87700-4\_50.

Rosenbrock, H. H. 1960. “An Automatic Method for Finding the Greatest or Least Value of a Function.” The Computer Journal 3 (3): 175–84. https://doi.org/10.1093/comjnl/3.3.175.

Schwefel, Hans-Paul. 1981. Numerical Optimization of Computer Models. John Wiley & Sons, Inc.