How to plot multi-objectives pareto frontier with DEAP in Python

Question

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.

Answer 1

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).

Answer 2

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.