makeZDT4Function function

ZDT4 Function

Builds and returns the two-objective ZDT4 test problem. For $m$ objective it is defined as follows [REMOVE_ME]$f(\mathbf{x}) = \left(f_1(\mathbf{x}_1), f_2(\mathbf{x})\right) [REMOVE_ME_2]$

with [REMOVE_ME]$f_1(\mathbf{x}_1) = \mathbf{x}_1, f_2(\mathbf{x}) = g(\mathbf{x}) h(f_1(\mathbf{x}_1), g(\mathbf{x})) [REMOVE_ME_2]$

where [REMOVE_ME]$g(\mathbf{x}) = 1 + 10 (m - 1) + \sum_{i = 2}^{m} (\mathbf{x}_i^2 - 10\cos(4\pi\mathbf{x}_i)), h(f_1, g) = 1 - \sqrt{\frac{f_1(\mathbf{x})}{g(\mathbf{x})}} [REMOVE_ME_2]$

and $\mathbf{x}_i \in [0,1], i = 1, \ldots, m$. This function has many Pareto-optimal fronts and is thus suited to test the algorithms ability to tackle multimodal problems.

makeZDT4Function(dimensions)

Arguments

• dimensions: [integer(1)]

  Number of decision variables.

Returns

[smoof_multi_objective_function]

Description

Builds and returns the two-objective ZDT4 test problem. For $m$ objective it is defined as follows

with

where

and $\mathbf{x}_i \in [0,1], i = 1, \ldots, m$. This function has many Pareto-optimal fronts and is thus suited to test the algorithms ability to tackle multimodal problems.

References

E. Zitzler, K. Deb, and L. Thiele. Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation, 8(2):173-195, 2000