smoof (version 1.6.0.3)

makeDTLZ5Function: DTLZ5 Function (family)

Description

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 $f_3(\mathbf{x})=(1+g({\mathbf{x}}_M))\cos ({\theta }_1\pi /2)\cos ({\theta }_2\pi /2)\cdots \sin ({\theta }_{M-2}\pi /2),$

$⋮$

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 $0\le x_i\le 1$, 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 $g({\mathbf{x}}_M)=\sum _{x_i\in {\mathbf{x}}_M}(x_i-0.5{)}^2$

Usage

makeDTLZ5Function(dimensions, n.objectives)

Value

[smoof_multi_objective_function]

Arguments

dimensions

[integer(1)]
Number of decision variables.

n.objectives

[integer(1)]
Number of objectives.

References

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