Builds and returns the multi-objective DTLZ4 test problem. It is a slight modification of the DTLZ2 problems by introducing the parameter \(\alpha\). The parameter is used to map \(\mathbf{x}_i \rightarrow \mathbf{x}_i^{\alpha}\).
The DTLZ4 test problem is defined as follows:
Minimize \(f_1(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \cos(x_2^\alpha\pi/2) \cdots \cos(x_{M-2}^\alpha\pi/2) \cos(x_{M-1}^\alpha\pi/2),\)
Minimize \(f_2(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \cos(x_2^\alpha\pi/2) \cdots \cos(x_{M-2}^\alpha\pi/2) \sin(x_{M-1}^\alpha\pi/2),\)
Minimize \(f_3(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \cos(x_2^\alpha\pi/2) \cdots \sin(x_{M-2}^\alpha\pi/2),\)
\(\vdots\\\)
Minimize \(f_{M-1}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1^\alpha\pi/2) \sin(x_2^\alpha\pi/2),\)
Minimize \(f_{M}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \sin(x_1^\alpha\pi/2),\)
with \(0 \leq x_i \leq 1\), for \(i=1,2,\dots,n,\)
where \(g(\mathbf{x}_M) = \sum\limits_{x_i\in \mathbf{x}_M}(x_i-0.5)^2\)
makeDTLZ4Function(dimensions, n.objectives, alpha = 100)[smoof_multi_objective_function]
[integer(1)]
Number of decision variables.
[integer(1)]
Number of objectives.
[numeric(1)]
Optional parameter. Default is 100, which is recommended by Deb et al.
K. Deb and L. Thiele and M. Laumanns and E. Zitzler. Scalable Multi-Objective Optimization Test Problems. Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zurich, 112, 2001