NSGA-III: Nondominated sorting genetic algorithm III

Reference: K. Deb and H. Jain, An evolutionary many-objective optimization algorithm using reference-point based non-dominated sorting approach, part I: Solving problems with box constraints, IEEE Transactions on Evolutionary Computation, 2014, 18(4): 577-601.

NSGA-III is an improved version of the classic multi-objective evolutionary algorithm (MOEA) NSGA-II. NSGA-III implements reference points to tackle the difficulties of many-objective optimization.

Variables	Meaning
npop	Population size
iter	Iteration number
lb	Lower bound
ub	Upper bound
nobj	The dimension of objective space
pc	Crossover probability (default = 1)
pm	Mutation probability (default = 1)
eta_c	Spread factor distribution index (default = 30)
eta_m	Perturbance factor distribution index (default = 20)
nvar	The dimension of decision space
pop	Population
objs	Objectives
V	Reference vectors
zmin	Ideal points
rank	Pareto rank
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(91, 400, np.array([0] * 7), np.array([1] * 7))

Output: