Xavier-MaYiMing / AnD

A many-objective evolutionary algorithm with angle-based selection and shift-based density estimation

AnD: A many-objective evolutionary algorithm with angle-based selection and shift-based density estimation

Reference: Liu Z Z, Wang Y, Huang P Q. AnD: A many-objective evolutionary algorithm with angle-based selection and shift-based density estimation[J]. Information Sciences, 2020, 509: 400-419.

AnD is a many-objective evolutionary algorithm (MaOEA).

Variables	Meaning
npop	Population size
iter	Iteration number
lb	Lower bound
ub	Upper bound
nobj	The dimension of objective space (default = 3)
pc	Crossover probability (default = 1)
pm	Mutation probability (default = 1)
eta_c	Spread factor distribution index (default = 20)
eta_m	Perturbance factor distribution index (default = 20)
nvar	The dimension of decision space
pop	Population
objs	Objectives
zmin	Ideal point
zmax	Nadir point
angle	The angle between each pair of objectives
remain	The remaining individuals to the next generation
pf	Pareto front

Test problem: DTLZ1

$$ \begin{aligned} & k = nvar - nobj + 1, \text{ the last $k$ variables is represented as $x_M$} \\ & g(x_M) = 100 \left[|x_M| + \sum_{x_i \in x_M}(x_i - 0.5)^2 - \cos(20\pi(x_i - 0.5)) \right] \\ & \min \\ & f_1(x) = \frac{1}{2}x_1x_2 \cdots x_{M - 1}(1 + g(x_M)) \\ & f_2(x) = \frac{1}{2}x_1x_2 \cdots (1 - x_{M - 1})(1 + g(x_M)) \\ & \vdots \\ & f_{M - 1}(x) = \frac{1}{2}x_1(1 - x_2)(1 + g(x_M)) \\ & f_M(x) = \frac{1}{2}(1 - x_1)(1 + g(x_M)) \\ & \text{subject to} \\ & x_i \in [0, 1], \quad \forall i = 1, \cdots, n \end{aligned} $$

Example

if __name__ == '__main__':
    main(100, 400, np.array([0] * 7), np.array([1] * 7))

Output: