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. $$f(\mathbf{x}) = \left(f_1(\mathbf{x}), f_2(\mathbf{x}, f_3(\mathbf{x}\right)$$ 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 \(-3 \leq \mathbf{x}_1, \mathbf{x}_2 \leq 3\).
makeViennetFunction()[smoof_multi_objective_function]
Viennet, R. (1996). Multicriteria optimization using a genetic algorithm for determining the Pareto set. International Journal of Systems Science 27 (2), 255-260.