\displaystyle 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}
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:
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.
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: 8 instance_indices: 1-10 dimensions: 2,20"
)
problem = suite[0]
print(problem.info) bbob_f008_i01_d02: a 2-dimensional single-objective problem (problem 105 of suite "b'bbob'" with name "BBOB suite problem f8 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=8,
dimension=14,
instance=42
)
problem(14 * [0])44934.816018537196You can find the C implementation of the problem here.