Detailed description of parameter adjustment for multi-objective genetic algorithm implemented by pymoo package NSGA2 algorithm
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- 1. Define the problem to be solved
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- 1.0 Parameter description of definition problem
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- 1.0.0 The solution problem must be set in the “`def _evaluate(self, x, out, *args, **kwargs)“` function
- 1.0.1 The problem must be wrapped with out[“F”] = [f1, f2]
- 1.0.2 Constraints must also be wrapped with out[“G”] = [g3]
- 1.0.3 “`def __init__(self):“` requires the following parameters to be defined
- 1.0.4 The constraint g is stated in the form of an inequality and will be selected according to the value less than or equal to 0.
- 2. Call the NSGA2 algorithm package to set parameters
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- 2.1 Parameter settings of NSGA2 function
- 3. Define the number of iterations 90 times
- 4. Solve the parameter x vector of the most Pareto optimal solution set
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- 4.2 Check the output solution of X
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- 5. Optimal solution set distribution of X vector parameters of Pareto optimal solution set
- 6. Draw the Pareto front
1. Define the problem to be solved
1.0 Parameter Description of Definition Questions
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1.0.0 The solution problem must be set in the
def _evaluate(self, x, out, *args, **kwargs)function -
1.0.1 questions must be wrapped with out[“F”] = [f1, f2]
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1.0.2 constraints must also be wrapped with out[“G”] = [g3]
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1.0.3
def __init__(self):The following parameters need to be defined n_vardefines what is to be solvedX
X
X number of variables
n_objdefines what is to be solvedf
f
f number of questions
n_ieq_constrdefines the number of constraintsxldefines what is to be solvedX
X
Lower limit of X parameter
xudefines what is to be solvedX
X
Upper limit of X parameter
f1,f2Definition problemg1,g2Define constraints-
1.0.4 The g of the constraint condition is stated in the form of an inequality and will be selected according to the value less than or equal to 0
import numpy as np
from pymoo.core.problem import ElementwiseProblem
class MyProblem(ElementwiseProblem):
def __init__(self):
super().__init__(n_var=2, # X number of variables
n_obj=2, # f number of questions
n_ieq_constr=1,# g number of constraints
xl=np.array([-2,-2]), # X lower limit of argument
xu=np.array([2,2])# X upper limit of argument
)
def _evaluate(self, x, out, *args, **kwargs):
# Function to be solved
f1 = np.cos(x[0] + x[1]) #100 * (x[0]**2 + x[1]**2)
f2 = np.sin(x[0] + x[1]) #(x[0]-1)**2 + x[1]**2
# f3 = (abs(x[0])<0.3) + (abs(x[1])<0.5)
# Constraints will select choices <= 0
# g1 = 2*(x[0]-0.1) * (x[0]-0.9) / 0.18
# g2 = - 20*(x[0]-0.4) * (x[0]-0.6) / 4.8
# g3 = ((x[0]**2)<0.5) + ((x[1]**2)>0.3)
g3 = x[0]-0.7 # + (abs(x[1])<0.3)
out["F"] = [f1, f2] #Problem to be solved
#out["G"] = [g1, g2,g3] #Constraints
out["G"] = [g3] #Constraints
problem = MyProblem()
2. Call the NSGA2 algorithm package to set parameters
2.1 Parameter settings of NSGA2 function
pop_sizezPopulation sizen_offspringsThe number of each generationsampling#Sampling settingscrossove()Crossover pairing settings-
probProbability settings for cross-matching
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eta
mutation()Mutation probability-
probis the probability setting of mutation
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eta
eliminate_duplicatesWe enable duplicate checking (“eliminate_duplicates=True”) to ensure that the offspring produced by mating differ from both itself and the existing population in terms of design space values.
from pymoo.algorithms.moo.nsga2 import NSGA2
from pymoo.operators.crossover.sbx import SBX
from pymoo.operators.mutation.pm import PM
from pymoo.operators.sampling.rnd import FloatRandomSampling,IntegerRandomSampling,BinaryRandomSampling
algorithm = NSGA2(
pop_size=90, # z population size
n_offsprings=100, #The number of each generation
sampling= FloatRandomSampling(), #sampling settings
#crossmatch
crossover=SBX(prob=0.9 #cross-pairing probability
, eta=15), #pairing efficiency
#Mutations
mutation=PM(prob=0.8 #Compilation probability
,eta=20),# Pairing efficiency
eliminate_duplicates=True
)
3. Define the number of iterations 90 times
from pymoo.termination import get_termination
termination = get_termination("n_gen", 90)
4. Solve the parameter x vector of the most Pareto optimal solution set
from pymoo.optimize import minimize
res = minimize(problem,
algorithm,
termination,
seed=1,
save_history=True,
verbose=True)
X = res.X # Solved parameters
F = res.F # Pareto optimal solution set
4.2 Check the output solution of X
array([[-1.22983903, -0.3408983 ],
[-1.17312926, -1.9683635],
[-1.62815244, -1.25867184],
[-1.59459202, -1.24720921],
[-0.85812605, -0.86104326],
[0.14217774, -1.79408273],
[-1.16038493, -0.45298884],
[-1.30857014, -0.53215836],
[-0.99480251, -1.0484723],
[-1.17506923, -0.83512127],
[-1.12330204, -1.13340585],
[-1.02395611, -1.33764674],
[-0.99658648, -0.88776711],
[-0.87963539, -0.86581268],
[-1.59330301, -0.09593218],
[-1.89860429, -1.07240541],
[-0.92025241, -0.88515559],
[-1.89221588, -1.02590525],
[-1.15977198, -0.61984559],
[-1.23391136, -1.89214062],
[-1.08575639, -1.1960931],
[-1.8422881 , -1.17730121],
[-1.97907088, -0.67847822],
[-1.19339619, -1.30837703],
[-1.81657534, -0.6468284],
[-1.34872892, -1.1691978],
[-1.70461135, -1.08101794],
[-1.28766298, -0.92085304],
[-1.10488217, -1.16702018],
[-1.45199598, -0.92807938],
[-1.83785271, -0.26933177],
[-1.10292853, -1.0760453],
[-1.97460715, -1.02344251],
[-1.92346673, -1.17730121],
[-1.35716933, -0.9513154],
[-1.07370789, -1.16339584],
[-1.61844778, -0.54832033],
[-1.69262569, -1.29666833],
[-1.1205858 , -1.9683635 ],
[-1.32886108, -1.09105746],
[-0.87963539, -0.85896725],
[-1.17319829, -1.50153054],
[-1.63954555, -1.28599005],
[-0.92662448, -0.93538073],
[-1.29072744, -0.82715754],
[-1.72496415, -1.22313643],
[-1.70410919, -1.36171497],
[-1.57300848, -1.04123091],
[-1.81522276, -0.66364657],
[-1.28643454, -1.14856238],
[-1.13870379, -0.8286136],
[-1.60254074, -1.21320856],
[-1.3972806 , -0.68146238],
[-1.37242908, -0.92807938],
[-1.29950364, -0.37689045],
[-1.32237812, -1.09105746],
[-1.59549137, -1.35399596],
[-0.86920703, -1.22313643],
[-1.38180886, -1.34157915],
[-1.46024398, -1.24232167],
[-1.12485534, -1.47579521],
[-1.24917941, -1.2408934],
[-1.6174287 , -1.02798238],
[-1.46214609, -0.68146238],
[-0.89315598, -0.95504252],
[-1.2693953 , -1.07649403],
[-1.31640451, -1.32237493],
[-1.2414329 , -1.15952844],
[-1.10828403, -0.80474544],
[-1.06864911, -0.83165391],
[-1.83785271, -1.1960931],
[-1.03382957, -1.50125804],
[-1.81678927, -0.71106355],
[-1.12485534, -1.50226124],
[-1.14170746, -1.05251568],
[-0.37583973, -1.94856256],
[-1.19888652, -0.86892885],
[-1.44462396, -0.94172587],
[-1.57293402, -1.19861388],
[-1.7873066 , -1.04123091],
[-1.19339619, -0.74337764],
[-1.41439116, -0.77744839],
[-1.03394747, -1.65557748],
[-1.29621172, -0.30606688],
[-0.85812605, -1.09549174],
[-1.31640451, -1.39906326],
[-1.36337969, -1.03256822],
[-1.59459202, -1.20585082],
[-1.10292853, -1.02523266],
[-1.85491578, -0.88327578]])
5. Optimal solution set distribution of X vector parameters of Pareto optimal solution set
import matplotlib.pyplot as plt plt.figure(figsize=(16,16)) plt.scatter(X[:,0],X[:,-1])
6. Draw the Pareto front
import matplotlib.pyplot as plt
plt.figure(figsize=(16,9))
plt.scatter(F[:,0],F[:,-1])
plt.savefig("NSGA2demo Pareto front.png")
