\displaystyle 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
Used symbols and definitions of, e.g., auxiliary functions are given in the following. Vectors are typeset in bold and refer to column vectors.
\otimes indicates element-wise multiplication of two D-dimensional vectors, \otimes:\mathcal{R}^D\times\mathcal{R}^D\to\mathcal{R}^D, (\mathbf{x},\mathbf{y})\mapsto\mathrm{{diag}}(\mathbf{x})\times\mathbf{y}=(x_i\times y_i)_{i=1,\dots,D}
\|.\| denotes the Euclidean norm, \|\mathbf{x}\|^2=\sum_i x_i^2.
[.] denotes the nearest integer value
\mathbf{0} =(0,\dots,0)^{\mathrm{T}} all zero vector
\mathbf{1} =(1,\dots,1)^{\mathrm{T}} all one vector
\Lambda^{\!\alpha} is a diagonal matrix in D dimensions with the ith diagonal element as \lambda_{ii} = \alpha^{\frac{1}{2}\frac{i-1}{D-1}}, for i=1,\dots,D.
f^{{}}_\mathrm{pen} :\mathcal{R}^D\to\mathcal{R}, \mathbf{x}\mapsto\sum_{i=1}^D\max(0,|x_i| - 5)^2
\mathbf{1}_-^+ a D-dimensional vector with entries of -1 or 1 with equal probability independently drawn.
\mathbf{Q}, \mathbf{R} orthogonal (rotation) matrices. For one function in one dimension a different realization for respectively \mathbf{Q} and \mathbf{R} is used for each instantiation of the function. Orthogonal matrices are generated from standard normally distributed entries by Gram-Schmidt orthonormalization. Columns and rows of an orthogonal matrix form an orthonormal basis.
\mathbf{R} see \mathbf{Q}
T^{{\beta}}_\mathrm{asy} :\mathcal{R}^D\to\mathcal{R}^D, x_i\mapsto \begin{cases} x_i^{1+\beta \frac{i-1}{D-1}\sqrt{x_i}} & \text{~if~} x_i>0\\ x_i & \text{~otherwise~} \end{cases}, for i=1,\dots,D. See Figure 1.
T_\mathrm{\hspace*{-0.01em}osz} :\mathcal{R}^n\to\mathcal{R}^n, for any positive integer n (n=1 and n=D are used in the following), maps element-wise x\mapsto\mathrm{{sign}}(x)\exp\left(\hat{x} + 0.049\left(\sin(c_1 \hat{x}) + \sin(c_2 \hat{x})\right)\right) with \hat{x}= \begin{cases} \log(|x|) & \text{if~} x\not=0 \\ 0 & \text{otherwise} \end{cases}, \mathrm{{sign}}(x) = \begin{cases} -1 & \text{if~} x < 0 \\ 0 & \text{if~} x = 0 \\ 1 & \text{otherwise} \end{cases}, c_1 = \begin{cases} 10 & \text{if~} x > 0\\ 5.5 & \text{otherwise} \end{cases} and c_2 = \begin{cases} 7.9 & \text{if~} x > 0\\ 3.1 & \text{otherwise} \end{cases}. See Figure 1.
\mathbf{x}^\mathrm{opt} optimal solution vector, such that f(\mathbf{x^\mathrm{opt}}) is minimal.
Properties:
Highly multimodal function based on (Lunacek et al. 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.
Information gained from this function: Can the search behavior be local on the global scale but global on a local scale?
Click on a plot to see the same visualization for other problem instances.
To limit benchmarking experiments only to the first ten instances of this problem from 2009 in dimensions 2 and 20, instantiate the Suite as follows:
import cocoex
suite = cocoex.Suite(
suite_name="bbob",
suite_instance="year: 2009",
suite_options="function_indices: 24 instance_indices: 1-10 dimensions: 2,20"
)
problem = suite[0]
print(problem.info) bbob_f024_i01_d02: a 2-dimensional single-objective problem (problem 345 of suite "b'bbob'" with name "BBOB suite problem f24 instance 1 in 2D")Sometimes it can be useful to access a bbob problem without using it in a benchmarking experiment. For such needs, the problem can be instantiated for any instance number and dimension using the BareProblem interface. Note that the bare problem cannot be observed/logged.
import cocoex
problem = cocoex.BareProblem(
suite_name="bbob",
function=24,
dimension=14,
instance=42
)
problem(14 * [0])259.11113136127733You can find the C implementation of the problem here.