For the project I was working on, I set 3 different objectives as optimization target in DEAP, an evolution framework based on Python.
It can cope with multi-objectives problem using algorithm like NSGA-II. Is there anyway to generate the pareto frontier surface for the visualizing the results.
Following a recipe in this link (not my own) to calculate the Pareto Points you could do:
def simple_cull(inputPoints, dominates):
paretoPoints = set()
candidateRowNr = 0
dominatedPoints = set()
while True:
candidateRow = inputPoints[candidateRowNr]
inputPoints.remove(candidateRow)
rowNr = 0
nonDominated = True
while len(inputPoints) != 0 and rowNr < len(inputPoints):
row = inputPoints[rowNr]
if dominates(candidateRow, row):
# If it is worse on all features remove the row from the array
inputPoints.remove(row)
dominatedPoints.add(tuple(row))
elif dominates(row, candidateRow):
nonDominated = False
dominatedPoints.add(tuple(candidateRow))
rowNr += 1
else:
rowNr += 1
if nonDominated:
# add the non-dominated point to the Pareto frontier
paretoPoints.add(tuple(candidateRow))
if len(inputPoints) == 0:
break
return paretoPoints, dominatedPoints
def dominates(row, candidateRow):
return sum([row[x] >= candidateRow[x] for x in range(len(row))]) == len(row)
import random
inputPoints = [[random.randint(70,100) for i in range(3)] for j in range(500)]
paretoPoints, dominatedPoints = simple_cull(inputPoints, dominates)
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
dp = np.array(list(dominatedPoints))
pp = np.array(list(paretoPoints))
print(pp.shape,dp.shape)
ax.scatter(dp[:,0],dp[:,1],dp[:,2])
ax.scatter(pp[:,0],pp[:,1],pp[:,2],color='red')
import matplotlib.tri as mtri
triang = mtri.Triangulation(pp[:,0],pp[:,1])
ax.plot_trisurf(triang,pp[:,2],color='red')
plt.show()
, you will notice that the last part is applying a triangulation to the Pareto points and plotting it as a triangular surface. The results is this (where the red shape is the Pareto front):
EDIT: Also you might want to take a look at this (although it seems to be for 2D spaces).
Also you might want to take a look at this Link, which implements a more efficient way to solve for 2D pareto frontiers via block nested loop (BNL). This is 30 times faster than the brute force method above.
