\displaystyle 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.
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:
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).
Information gained from this function:
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: 21 instance_indices: 1-10 dimensions: 2,20"
)
problem = suite[0]
print(problem.info) bbob_f021_i01_d02: a 2-dimensional single-objective problem (problem 300 of suite "b'bbob'" with name "BBOB suite problem f21 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=21,
dimension=14,
instance=42
)
problem(14 * [0])204.18864112542298You can find the C implementation of the problem here.