Builds and returns the multi-objective ED2 test problem.
The ED2 test problem is defined as follows:
Minimize \(f_j(\mathbf{x}) = \frac{1}{F_{natmin}(\mathbf{x}) + 1} \cdot \tilde{p}(\Theta (\mathbf{X}))\), for \(j = 1, \ldots, m\),
with \(\mathbf{x} = (x_1, \ldots, x_n)^T\), where \(0 \leq x_i \leq 1\), and \(\Theta = (\theta_1, \ldots, \theta_{m-1})\), where \(0 \le \theta_j \le \frac{\pi}{2}\), for \(i = 1, \ldots, n,\) and \(j = 1, \ldots, m - 1\).
Moreover \(F_{natmin}(\mathbf{x}) = b + (r(\mathbf{x}) - a) + 0.5 + 0.5 \cdot (2 \pi \cdot (r(\mathbf{x}) - a) + \pi)\)
with \(a \approx 0.051373\), \(b \approx 0.0253235\), and \(r(\mathbf{X}) = \sqrt{x_m^2 + \ldots, x_n^2}\), as well as
\(\tilde{p}_1(\Theta) = \cos(\theta_1)^{2/\gamma}\),
\(\tilde{p}_j(\Theta) = \left( \sin(\theta_1) \cdot \ldots \cdot \sin(\theta_{j - 1}) \cdot \cos(\theta_j) \right)^{2/\gamma}\), for \(2 \le j \le m - 1\),
and \(\tilde{p}_m(\Theta) = \left( \sin(\theta_1) \cdot \ldots \cdot \sin(\theta_{m - 1}) \right)^{2/\gamma}\).
makeED2Function(dimensions, n.objectives, gamma = 2, theta)[smoof_multi_objective_function]
[integer(1)]
Number of decision variables.
[integer(1)]
Number of objectives.
[numeric(1)]
Optional parameter. Default is 2, which is recommended by Emmerich and Deutz.
[numeric(dimensions)]
Parameter vector, whose components have to be between 0 and 0.5*pi.
The default is theta = (pi/2) * x (with x being the point from the decision space) as recommended by Emmerich and Deutz.
M. T. M. Emmerich and A. H. Deutz. Test Problems based on Lame Superspheres. Proceedings of the International Conference on Evolutionary Multi-Criterion Optimization (EMO 2007), pp. 922-936, Springer, 2007.