Description

The Viennet test problem VNT is designed for three objectives only. It has a discrete set of Pareto fronts. It is defined by the following formulae.

f(\mathbf{x}) = ≤ft(f_1(\mathbf{x}), f_2(\mathbf{x}, f_3(\mathbf{x}\right)

with

f_1(\mathbf{x}) = 0.5(\mathbf{x}_1^2 + \mathbf{x}_2^2) + \sin(\mathbf{x}_1^2 + \mathbf{x}_2^2)

f_2(\mathbf{x}) = \frac{(3\mathbf{x}_1 + 2\mathbf{x}_2 + 4)^2}{8} + \frac{(\mathbf{x}_1 - \mathbf{x}_2 + 1)^2}{27} + 15

f_3(\mathbf{x}) = \frac{1}{\mathbf{x}_1^2 + \mathbf{x}_2^2 + 1} - 1.1\exp(-(\mathbf{x}_1^1 + \mathbf{x}_2^2))

with box constraints -3 ≤q \mathbf{x}_1, \mathbf{x}_2 ≤q 3.

Usage

Value

[smoof_multi_objective_function]

References

Viennet, R. (1996). Multicriteria optimization using a genetic algorithm for determining the Pareto set. International Journal of Systems Science 27 (2), 255-260.