makeDTLZ5Function: DTLZ5 Function (family)

In jakobbossek/smoof: Single and Multi-Objective Optimization Test Functions

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 sub-populations 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

Usage

makeDTLZ5Function(dimensions, n.objectives)

Arguments

dimensions	[integer(1)] Number of decision variables.
n.objectives	[integer(1)] Number of objectives.

Value

[smoof_multi_objective_function] Returns an instance of the DTLZ5 family as a smoof_multi_objective_function object.

Note

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.

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

jakobbossek/smoof documentation built on Feb. 17, 2024, 2:23 a.m.