MOEA/D-IEpsilon: Multi-objective evolutionary algorithm based on decomposition and improved epsilon constraint-handling mechanism

Reference: Fan Z, Li W, Cai X, et al. An improved epsilon constraint-handling method in MOEA/D for CMOPs with large infeasible regions[J]. Soft Computing, 2019, 23: 12491-12510.

MOEA/D-IEpsilon is a constrained multi-objective evolutionary algorithm (CMOEA). The epsilon level is adjust according to the ratio of feasible to total solutions in the current population.

Variables	Meaning
npop	Population size
iter	Iteration number
lb	Lower bound
ub	Upper bound
T	Neighborhood size (default = 30)
delta	The probability of selecting individuals in the neighborhood (default = 0.9)
nr	The maximal number of solutions replaced by a child (default = 2)
tau	Control the scale factor multiplied by the maximum overall constraint violation (default = 0.1)
alpha	Control the searching preference between the feasible and infeasible regions (default = 0.95)
Tc	Control generation (default = 0.8 * iter)
CR	Crossover rate (default = 1)
F	Mutation scalar number (default = 0.5)
pm	Mutation probability (default = 1)
eta_m	Spread factor distribution index (default = 20)
nvar	The dimension of decision space
nobj	The dimension of objective space
V	Weight vectors
B	The T closet weight vectors
pop	Population
objs	Objectives
phi	Constraint violations
phi_max	The maximum overall constraint violation found so far
rk	The ratio of feasible solutions to all solutions
z	Ideal point
NS	Feasible non-dominated solutions
rp	Random permutation
off	Offspring
off_obj	Offspring objecrtive
off_phi	Offspring constraint violation
c	Update counter

Test problem: LIR-CMOP6

$$ \begin{aligned} & J_1={3, 5, \cdots, 29}, J_2 = {2, 4, \cdots, 30} \\ & g_1(x) = \sum_{i \in J_1} (x_i - \sin(0.5i\pi x_1/30))^2 \\ & g_2(x) = \sum_{i \in J_2} (x_i - \cos(0.5i\pi x_1/30))^2 \\ &\min \\ & f_1(x) = x_1 + 10g_1(x) + 0.7057 \\ & f_2(x) = 1 - x_1^2 + 10g_2(x) + 0.7057 \\ & \text{subject to} \\ & c_k(x) = ((f_1 - p_k)\cos\theta - (f_2 - q_k)\sin\theta)^2/a_k^2 + ((f_1 - p_k)\sin\theta - (f_2 - q_k)\cos\theta)^2/b_k^2 \geq r \\ & p_k = [1.8, 2.8], \quad q_k = [1.8, 2.8], \quad a_k = [2, 2], \quad b_k = [8, 8], \quad k=1, 2 \\ & r = 0.1, \quad \theta = -0.25 \pi \\ & x_i \in [0, 1], \quad i = 1, \cdots, 30 \end{aligned} $$

Example

if __name__ == '__main__':
    main(300, 1000, np.array([0] * 30), np.array([1] * 30))

Output: