Builds and returns the multi-objective DTLZ3 test problem. The formula is very similar to the formula of DTLZ2, but it uses the \(g\) function of DTLZ1, which introduces a lot of local Pareto-optimal fronts. Thus, this problems is well suited to check the ability of an optimizer to converge to the global Pareto-optimal front.
The DTLZ3 test problem is defined as follows:
Minimize \(f_1(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1\pi/2) \cos(x_2\pi/2) \cdots \cos(x_{M-2}\pi/2) \cos(x_{M-1}\pi/2),\)
Minimize \(f_2(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1\pi/2) \cos(x_2\pi/2) \cdots \cos(x_{M-2}\pi/2) \sin(x_{M-1}\pi/2),\)
Minimize \(f_3(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1\pi/2) \cos(x_2\pi/2) \cdots \sin(x_{M-2}\pi/2),\)
\(\vdots\\\)
Minimize \(f_{M-1}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \cos(x_1\pi/2) \sin(x_2\pi/2),\)
Minimize \(f_{M}(\mathbf{x}) = (1+g(\mathbf{x}_M)) \sin(x_1\pi/2),\)
with \(0 \leq x_i \leq 1\), for \(i=1,2,\dots,n,\)
where \(g(\mathbf{x}_M) = 100 \left[|\mathbf{x}_M| + \sum\limits_{x_i \in \mathbf{x}_M} (x_i - 0.5)^2 - \cos(20\pi(x_i - 0.5))\right]\)
makeDTLZ3Function(dimensions, n.objectives)[smoof_multi_objective_function]
[integer(1)]
Number of decision variables.
[integer(1)]
Number of objectives.
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