makeDTLZ3Function {smoof}

DTLZ3 Function (family)

Description

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]

Usage

makeDTLZ3Function(dimensions, n.objectives)

Arguments

dimensions

[integer(1)]
Number of decision variables.

n.objectives

[integer(1)]
Number of objectives.

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