makeZDT3Function function

ZDT3 Function

Builds and returns the two-objective ZDT3 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]$g(\mathbf{x}) = 1 + \frac{9}{m - 1} \sum_{i = 2}^m \mathbf{x}_i, h(f_1, g) = 1 - \sqrt{\frac{f_1(\mathbf{x})}{g(\mathbf{x})}} - \left(\frac{f_1(\mathbf{x})}{g(\mathbf{x})}\right)\sin(10\pi f_1(\mathbf{x})) [REMOVE_ME_2]$

and $\mathbf{x}_i \in [0,1], i = 1, \ldots, m$. This function has some discontinuities in the Pareto-optimal front introduced by the sine term in the $h$ function (see above). The front consists of multiple convex parts.

makeZDT3Function(dimensions)

Arguments

• dimensions: [integer(1)]

  Number of decision variables.

Returns

[smoof_multi_objective_function]

Description

Builds and returns the two-objective ZDT3 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 some discontinuities in the Pareto-optimal front introduced by the sine term in the $h$ function (see above). The front consists of multiple convex parts.

References

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