Builds and returns the multi-objective DTLZ5 test problem. This problem can be characterized by a disconnected Pareto-optimal front in the search space. This introduces a new challenge to evolutionary multi-objective optimizers, i.e., to maintain different subpopulations within the search space to cover the entire Pareto-optimal front.
The DTLZ5 test problem is defined as follows:
Minimize \(f_1(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(\theta_1\pi/2) \cos(\theta_2\pi/2) \cdots \cos(\theta_{M-2}\pi/2) \cos(\theta_{M-1}\pi/2),\)
Minimize \(f_2(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(\theta_1\pi/2) \cos(\theta_2\pi/2) \cdots \cos(\theta_{M-2}\pi/2) \sin(\theta_{M-1}\pi/2),\)
Minimize \(f_3(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(\theta_1\pi/2) \cos(\theta_2\pi/2) \cdots \sin(\theta_{M-2}\pi/2),\)
\(\vdots\\\)
Minimize \(f_{M-1}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(\theta_1\pi/2) \sin(\theta_2\pi/2),\)
Minimize \(f_{M}((1+g(\mathbf{x}_M)) \sin(\theta_1\pi/2),\)
with \(0 \leq x_i \leq 1\), for \(i=1,2,\dots,n,\)
where \(\theta_i = \frac{\pi}{4(1+ g(\mathbf{x}_M))} (1+2g(\mathbf{x}_M)x_i),\) for \(i = 2,3,\dots,(M-1)\)
and \(g(\mathbf{x}_M) = \sum\limits_{x_i\in\mathbf{x}_M}(x_i-0.5)^2\)
makeDTLZ5Function(dimensions, n.objectives)[integer(1)]
Number of decision variables.
[integer(1)]
Number of objectives.
[smoof_multi_objective_function]
This problem definition does not exist in the succeeding work of Deb et al. (K. Deb
and L. Thiele and M. Laumanns and E. Zitzler (2002). Scalable multi-objective optimization
test problems, Proceedings of the IEEE Congress on Evolutionary Computation, pp. 825-830).
Also, note that in case of a bi-objective scenario (n.objectives = 2L) DTLZ2 and DTLZ5 are identical.
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