CCMO: A coevolutionary constrained multiobjective optimization (CCMO) framework

Reference: Tian Y, Zhang T, Xiao J, et al. A coevolutionary framework for constrained multiobjective optimization problems[J]. IEEE Transactions on Evolutionary Computation, 2020, 25(1): 102-116.

The CCMO belongs to the category of multi-objective evolutionary algorithms (MOEAs). CCMO is a powerful algorithm to solve the constrained multiobjective optimization (CMO) problems.

Variables	Meaning
npop	Population size
iter	Iteration number
lb	Lower bound
ub	Upper bound
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
pop1	Population1
pop2	Population2
objs1	Objectives1
objs2	Objective2
CV1	Constraint violation1
CV2	Constraint violation2
F1	Fitness1
F2	Fitness2
mating_pool	Mating pool
offspring	Offspring
pf	Pareto front

Test problem: MW11

Reference: Ma Z, Wang Y. Evolutionary constrained multiobjective optimization: Test suite construction and performance comparisons[J]. IEEE Transactions on Evolutionary Computation, 2019, 23(6): 972-986.

Define $m$ is the dimension of objective space, and $n$ is the dimension of decistion space.

$$ \begin{aligned} &g3(x)=1+\sum_{i=m}^n2(x_i+(x_{i-1}-0.5)^2-1)^2\\ &f_1(x)=\sqrt{2}g_3x_1\\ &f_2(x)=g_3\sqrt{2-(f_1/g_3)^2}\\ &\text{subject to}\\ &c_1(x)=(3-f_1^2-f_2)(3-2f_1^2-f_2)\geq0\\ &c_2(x)=(3-0.625f_1^2-f_2)(3-7f_1^2-f_2)\leq0\\ &c_3(x)=(1.62-0.18f_1^2-f_2)(1.125-0.125f_1^2-f_2)\geq0\\ &c_4(x)=(2.07-0.23f_1^2-f_2)(0.63-0.07f_1^2-f_2)\leq0\\ &0\leq x_i\leq 1, \quad i=1,\cdots,n \end{aligned} $$

Example

if __name__ == '__main__':
    main(100, 500, np.array([0] * 15), np.array([1] * 15))

Output:

Iteration 20 completed.
Iteration 40 completed.
Iteration 60 completed.
Iteration 80 completed.
Iteration 100 completed.
Iteration 120 completed.
Iteration 140 completed.
Iteration 160 completed.
Iteration 180 completed.
Iteration 200 completed.
Iteration 220 completed.
Iteration 240 completed.
Iteration 260 completed.
Iteration 280 completed.
Iteration 300 completed.
Iteration 320 completed.
Iteration 340 completed.
Iteration 360 completed.
Iteration 380 completed.
Iteration 400 completed.
Iteration 420 completed.
Iteration 440 completed.
Iteration 460 completed.
Iteration 480 completed.
Iteration 500 completed.