NSGA-II: Non-dominated Sorting Genetic Algorithm¶

The algorithm is implemented based on [5]. The algorithm follows the general outline of a genetic algorithm with a modified mating and survival selection. In NSGA-II, first, individuals are selected frontwise. By doing so, there will be the situation where a front needs to be split because not all individuals are allowed to survive. In this splitting front, solutions are selected based on crowding distance.

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The crowding distance is the Manhatten Distance in the objective space. However, the extreme points are desired to be kept every generation and, therefore, get assigned a crowding distance of infinity.

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Furthermore, to increase some selection pressure, NSGA-II uses a binary tournament mating selection. Each individual is first compared by rank and then crowding distance. There is also a variant in the original C code where instead of using the rank, the domination criterion between two solutions is used.

Example¶

[1]:

from pymoo.algorithms.moo.nsga2 import NSGA2
from pymoo.problems import get_problem
from pymoo.optimize import minimize
from pymoo.visualization.scatter import Scatter

problem = get_problem("zdt1")

algorithm = NSGA2(pop_size=100)

res = minimize(problem,
               algorithm,
               ('n_gen', 200),
               seed=1,
               verbose=False)

plot = Scatter()
plot.add(problem.pareto_front(), plot_type="line", color="black", alpha=0.7)
plot.add(res.F, facecolor="none", edgecolor="red")
plot.show()

[1]:

<pymoo.visualization.scatter.Scatter at 0x11b307920>

../../_images/algorithms_moo_nsga2_10_1.png

Moreover, we can customize NSGA-II to solve a problem with binary decision variables, for example, ZDT5.

[2]:

from pymoo.algorithms.moo.nsga2 import NSGA2
from pymoo.problems import get_problem
from pymoo.operators.crossover.pntx import TwoPointCrossover
from pymoo.operators.mutation.bitflip import BitflipMutation
from pymoo.operators.sampling.rnd import BinaryRandomSampling
from pymoo.optimize import minimize
from pymoo.visualization.scatter import Scatter

problem = get_problem("zdt5")

algorithm = NSGA2(pop_size=100,
                  sampling=BinaryRandomSampling(),
                  crossover=TwoPointCrossover(),
                  mutation=BitflipMutation(),
                  eliminate_duplicates=True)

res = minimize(problem,
               algorithm,
               ('n_gen', 500),
               seed=1,
               verbose=False)

Scatter().add(res.F).show()

[2]:

<pymoo.visualization.scatter.Scatter at 0x147a52810>

../../_images/algorithms_moo_nsga2_12_1.png

API¶

class pymoo.algorithms.moo.nsga2.NSGA2(pop_size=100, sampling=<pymoo.operators.sampling.rnd.FloatRandomSampling object>, selection=<pymoo.operators.selection.tournament.TournamentSelection object>, crossover=<pymoo.operators.crossover.sbx.SBX object>, mutation=<pymoo.operators.mutation.pm.PM object>, survival=<pymoo.operators.survival.rank_and_crowding.classes.RankAndCrowding object>, output=<pymoo.util.display.multi.MultiObjectiveOutput object>, **kwargs)