makeDTLZ4Function: DTLZ4 Function (family)

In smoof: Single and Multi-Objective Optimization Test Functions

DTLZ4 Function (family)

Description

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

Usage

makeDTLZ4Function(dimensions, n.objectives, alpha = 100)

Arguments

dimensions	[integer(1)] Number of decision variables.
n.objectives	[integer(1)] Number of objectives.
alpha	[numeric(1)] Optional parameter. Default is 100, which is recommended by Deb et al.

Value

[smoof_multi_objective_function]

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

smoof documentation built on March 31, 2023, 11:48 p.m.