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NSGA_III.py
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NSGA_III.py
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#!/usr/bin/env python
# -*- coding: utf-8 -*-
# @Time : 2023/8/22 09:14
# @Author : Xavier Ma
# @Email : [email protected]
# @File : NSGA_III.py
# @Statement : Nondominated sorting genetic algorithm III (NSGA-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.
import numpy as np
import matplotlib.pyplot as plt
from collections import Counter
from itertools import combinations
from scipy.linalg import LinAlgError
from scipy.spatial.distance import cdist
def cal_obj(pop, nobj):
# DTLZ1
g = 100 * (pop.shape[1] - nobj + 1 + np.sum((pop[:, nobj - 1:] - 0.5) ** 2 - np.cos(20 * np.pi * (pop[:, nobj - 1:] - 0.5)), axis=1))
objs = np.zeros((pop.shape[0], nobj))
temp_pop = pop[:, : nobj - 1]
for i in range(nobj):
f = 0.5 * (1 + g)
f *= np.prod(temp_pop[:, : temp_pop.shape[1] - i], axis=1)
if i > 0:
f *= 1 - temp_pop[:, temp_pop.shape[1] - i]
objs[:, i] = f
return objs
def factorial(n):
# calculate n!
if n == 0 or n == 1:
return 1
else:
return n * factorial(n - 1)
def combination(n, m):
# choose m elements from an n-length set
if m == 0 or m == n:
return 1
elif m > n:
return 0
else:
return factorial(n) // (factorial(m) * factorial(n - m))
def reference_points(npop, nvar):
# calculate approximately npop uniformly distributed reference points on nvar dimensions
h1 = 0
while combination(h1 + nvar, nvar - 1) <= npop:
h1 += 1
points = np.array(list(combinations(np.arange(1, h1 + nvar), nvar - 1))) - np.arange(nvar - 1) - 1
points = (np.concatenate((points, np.zeros((points.shape[0], 1)) + h1), axis=1) - np.concatenate((np.zeros((points.shape[0], 1)), points), axis=1)) / h1
if h1 < nvar:
h2 = 0
while combination(h1 + nvar - 1, nvar - 1) + combination(h2 + nvar, nvar - 1) <= npop:
h2 += 1
if h2 > 0:
temp_points = np.array(list(combinations(np.arange(1, h2 + nvar), nvar - 1))) - np.arange(nvar - 1) - 1
temp_points = (np.concatenate((temp_points, np.zeros((temp_points.shape[0], 1)) + h2), axis=1) - np.concatenate((np.zeros((temp_points.shape[0], 1)), temp_points), axis=1)) / h2
temp_points = temp_points / 2 + 1 / (2 * nvar)
points = np.concatenate((points, temp_points), axis=0)
return points
def nd_sort(objs):
# fast non-domination sort
(npop, nobj) = objs.shape
n = np.zeros(npop, dtype=int) # the number of individuals that dominate this individual
s = [] # the index of individuals that dominated by this individual
rank = np.zeros(npop, dtype=int)
ind = 0
pfs = {ind: []} # Pareto fronts
for i in range(npop):
s.append([])
for j in range(npop):
if i != j:
less = equal = more = 0
for k in range(nobj):
if objs[i, k] < objs[j, k]:
less += 1
elif objs[i, k] == objs[j, k]:
equal += 1
else:
more += 1
if less == 0 and equal != nobj:
n[i] += 1
elif more == 0 and equal != nobj:
s[i].append(j)
if n[i] == 0:
pfs[ind].append(i)
rank[i] = ind
while pfs[ind]:
pfs[ind + 1] = []
for i in pfs[ind]:
for j in s[i]:
n[j] -= 1
if n[j] == 0:
pfs[ind + 1].append(j)
rank[j] = ind + 1
ind += 1
pfs.pop(ind)
return pfs, rank
def selection(pop, pc, rank, k=2):
# binary tournament selection
(npop, nvar) = pop.shape
nm = int(npop * pc)
nm = nm if nm % 2 == 0 else nm + 1
mating_pool = np.zeros((nm, nvar))
for i in range(nm):
[ind1, ind2] = np.random.choice(npop, k, replace=False)
if rank[ind1] <= rank[ind2]:
mating_pool[i] = pop[ind1]
else:
mating_pool[i] = pop[ind2]
return mating_pool
def crossover(mating_pool, lb, ub, pc, eta_c):
# simulated binary crossover (SBX)
(noff, nvar) = mating_pool.shape
nm = int(noff / 2)
parent1 = mating_pool[:nm]
parent2 = mating_pool[nm:]
beta = np.zeros((nm, nvar))
mu = np.random.random((nm, nvar))
flag1 = mu <= 0.5
flag2 = ~flag1
beta[flag1] = (2 * mu[flag1]) ** (1 / (eta_c + 1))
beta[flag2] = (2 - 2 * mu[flag2]) ** (-1 / (eta_c + 1))
beta = beta * (-1) ** np.random.randint(0, 2, (nm, nvar))
beta[np.random.random((nm, nvar)) < 0.5] = 1
beta[np.tile(np.random.random((nm, 1)) > pc, (1, nvar))] = 1
offspring1 = (parent1 + parent2) / 2 + beta * (parent1 - parent2) / 2
offspring2 = (parent1 + parent2) / 2 - beta * (parent1 - parent2) / 2
offspring = np.concatenate((offspring1, offspring2), axis=0)
offspring = np.min((offspring, np.tile(ub, (noff, 1))), axis=0)
offspring = np.max((offspring, np.tile(lb, (noff, 1))), axis=0)
return offspring
def mutation(pop, lb, ub, pm, eta_m):
# polynomial mutation
(npop, nvar) = pop.shape
lb = np.tile(lb, (npop, 1))
ub = np.tile(ub, (npop, 1))
site = np.random.random((npop, nvar)) < pm / nvar
mu = np.random.random((npop, nvar))
delta1 = (pop - lb) / (ub - lb)
delta2 = (ub - pop) / (ub - lb)
temp = np.logical_and(site, mu <= 0.5)
pop[temp] += (ub[temp] - lb[temp]) * ((2 * mu[temp] + (1 - 2 * mu[temp]) * (1 - delta1[temp]) ** (eta_m + 1)) ** (1 / (eta_m + 1)) - 1)
temp = np.logical_and(site, mu > 0.5)
pop[temp] += (ub[temp] - lb[temp]) * (1 - (2 * (1 - mu[temp]) + 2 * (mu[temp] - 0.5) * (1 - delta2[temp]) ** (eta_m + 1)) ** (1 / (eta_m + 1)))
pop = np.min((pop, ub), axis=0)
pop = np.max((pop, lb), axis=0)
return pop
def environmental_selection(pop, objs, zmin, npop, V):
# NSGA-III environmental selection
pfs, rank = nd_sort(objs)
nobj = objs.shape[1]
selected = np.full(pop.shape[0], False)
ind = 0
while np.sum(selected) + len(pfs[ind]) <= npop:
selected[pfs[ind]] = True
ind += 1
K = npop - np.sum(selected)
# select the remaining K solutions
objs1 = objs[selected]
objs2 = objs[pfs[ind]]
npop1 = objs1.shape[0]
npop2 = objs2.shape[0]
nv = V.shape[0]
temp_objs = np.concatenate((objs1, objs2), axis=0)
t_objs = temp_objs - zmin
# extreme points
extreme = np.zeros(nobj)
w = 1e-6 + np.eye(nobj)
for i in range(nobj):
extreme[i] = np.argmin(np.max(t_objs / w[i], axis=1))
# intercepts
try:
hyperplane = np.matmul(np.linalg.inv(t_objs[extreme.astype(int)]), np.ones((nobj, 1)))
if np.any(hyperplane == 0):
a = np.max(t_objs, axis=0)
else:
a = 1 / hyperplane
except LinAlgError:
a = np.max(t_objs, axis=0)
t_objs /= a.reshape(1, nobj)
# association
cosine = 1 - cdist(t_objs, V, 'cosine')
distance = np.sqrt(np.sum(t_objs ** 2, axis=1).reshape(npop1 + npop2, 1)) * np.sqrt(1 - cosine ** 2)
dis = np.min(distance, axis=1)
association = np.argmin(distance, axis=1)
temp_rho = dict(Counter(association[: npop1]))
rho = np.zeros(nv)
for key in temp_rho.keys():
rho[key] = temp_rho[key]
# selection
choose = np.full(npop2, False)
v_choose = np.full(nv, True)
while np.sum(choose) < K:
temp = np.where(v_choose)[0]
jmin = np.where(rho[temp] == np.min(rho[temp]))[0]
j = temp[np.random.choice(jmin)]
I = np.where(np.bitwise_and(~choose, association[npop1:] == j))[0]
if I.size > 0:
if rho[j] == 0:
s = np.argmin(dis[npop1 + I])
else:
s = np.random.randint(I.size)
choose[I[s]] = True
rho[j] += 1
else:
v_choose[j] = False
selected[np.array(pfs[ind])[choose]] = True
return pop[selected], objs[selected], rank[selected]
def main(npop, iter, lb, ub, nobj=3, pc=1, pm=1, eta_c=30, eta_m=20):
"""
The main function
:param npop: population size
:param iter: iteration number
:param lb: lower bound
:param ub: upper bound
:param nobj: the dimension of objective space
:param pc: crossover probability (default = 1)
:param pm: mutation probability (default = 1)
:param eta_c: spread factor distribution index (default = 30)
:param eta_m: perturbance factor distribution index (default = 20)
:return:
"""
# Step 1. Initialization
nvar = len(lb) # the dimension of decision space
pop = np.random.uniform(lb, ub, (npop, nvar)) # population
objs = cal_obj(pop, nobj) # objectives
V = reference_points(npop, nobj) # reference vectors
zmin = np.min(objs, axis=0) # ideal points
[pfs, rank] = nd_sort(objs) # Pareto rank
# Step 2. The main loop
for t in range(iter):
if (t + 1) % 50 == 0:
print('Iteration: ' + str(t + 1) + ' completed.')
# Step 2.1. Mating selection + crossover + mutation
mating_pool = selection(pop, pc, rank)
off = crossover(mating_pool, lb, ub, pc, eta_c)
off = mutation(off, lb, ub, pm, eta_m)
off_objs = cal_obj(off, nobj)
# Step 2.2. Environmental selection
zmin = np.min((zmin, np.min(off_objs, axis=0)), axis=0)
pop, objs, rank = environmental_selection(np.concatenate((pop, off), axis=0), np.concatenate((objs, off_objs), axis=0), zmin, npop, V)
# Step 3. Sort the results
pf = objs[rank == 0]
ax = plt.figure().add_subplot(111, projection='3d')
ax.view_init(45, 45)
x = [o[0] for o in pf]
y = [o[1] for o in pf]
z = [o[2] for o in pf]
ax.scatter(x, y, z, color='red')
ax.set_xlabel('objective 1')
ax.set_ylabel('objective 2')
ax.set_zlabel('objective 3')
plt.title('The Pareto front of DTLZ1')
plt.savefig('Pareto front')
plt.show()
if __name__ == '__main__':
main(91, 400, np.array([0] * 7), np.array([1] * 7))