Many-objective

Throughout this section, we will solve the three-objective problem DTLZ2 imported from pymoo.

The definition of “many-objective” differs according to the refence used. Some issues arise in maintaining population diversity in the survival stage for problems with more than two objectives. Therefore we will consider here “many-objective problems those with more than two objectives.

For more details about the algorithms used, please refer to the Algorithms section.

import matplotlib.pyplot as plt
from pymoo.optimize import minimize
from pymoo.util.ref_dirs import get_reference_directions

from pymoode.algorithms import GDE3, NSDER
from pymoode.survival import RankAndCrowding

DTLZ2

\[\begin{split}\begin{align} \text{min} \; \; & f_1(\boldsymbol{x}) = \mathrm{cos}(\pi / 2 x_1) \, \mathrm{cos}(\pi / 2 x_2) (1 + g(\boldsymbol{x}))\\ & f_2(\boldsymbol{x}) = \mathrm{cos}(\pi / 2 x_1) \, \mathrm{sin}(\pi / 2 x_2) (1 + g(\boldsymbol{x}))\\ & f_3(\boldsymbol{x}) = \mathrm{sin}(\pi / 2 x_1) (1 + g(\boldsymbol{x}))\\ \text{s.t.} \; \; & g(\boldsymbol{x}) = \sum_{i=1}^{3}(x_i - 0.5) ^ 2\\ & 0.0 \leq \boldsymbol{x} \leq 1.0 \end{align}\end{split}\]

from pymoo.problems import get_problem

problem = get_problem('dtlz2')

NGEN = 150
POPSIZE = 136
SEED = 5

GDE3-MNN

Let us instantiate a GDE3 algorithm and pass as the survival operator RankAndCrowding(crowding_func="mnn"), which is suitable for problems with more than two objectives. Alternatively, one could have directly imported the GDE3MNN algorithm.

from pymoode.algorithms import GDE3MNN

gde3 = GDE3(
    pop_size=POPSIZE,
    variant='DE/rand/1/bin',
    CR=0.2,
    F=(0.0, 1.0),
    gamma=1e-4,
    survival=RankAndCrowding(crowding_func='mnn'),
)

res_gde3 = minimize(
    problem,
    gde3,
    ('n_gen', NGEN),
    seed=SEED,
    save_history=False,
    verbose=False,
)

fig, ax = plt.subplots(figsize=[6, 5], dpi=70, subplot_kw={'projection': '3d'})

ax.scatter(
    problem.pareto_front()[:, 0],
    problem.pareto_front()[:, 1],
    problem.pareto_front()[:, 2],
    color='firebrick',
    label='True Front',
    marker='o',
)

ax.scatter(
    res_gde3.F[:, 0],
    res_gde3.F[:, 1],
    res_gde3.F[:, 2],
    color='navy',
    label='GDE3-MNN',
    marker='o',
)

ax.view_init(elev=30, azim=30)

ax.set_xlabel('$f_1$')
ax.set_ylabel('$f_2$')
ax.set_zlabel('$f_3$')
ax.legend()
fig.tight_layout()

NSDE-R

Let us now instantiate the NSDE-R algorithm, which uses DE reproduction operators with the survival strategy of NSGA-III.

ref_dirs = get_reference_directions('das-dennis', 3, n_partitions=15)

nsder = NSDER(
    ref_dirs=ref_dirs,
    pop_size=POPSIZE,
    variant='DE/rand/1/bin',
    CR=0.5,
    F=(0.0, 1.0),
    gamma=1e-4,
)

res_nsder = minimize(
    problem,
    nsder,
    ('n_gen', NGEN),
    seed=SEED,
    save_history=False,
    verbose=False,
)

fig, ax = plt.subplots(figsize=[6, 5], dpi=70, subplot_kw={'projection': '3d'})

ax.scatter(
    problem.pareto_front()[:, 0],
    problem.pareto_front()[:, 1],
    problem.pareto_front()[:, 2],
    color='firebrick',
    label='True Front',
    marker='o',
)

ax.scatter(
    res_nsder.F[:, 0],
    res_nsder.F[:, 1],
    res_nsder.F[:, 2],
    color='navy',
    label='NSDE-R',
    marker='o',
)

ax.view_init(elev=30, azim=30)

ax.set_xlabel('$f_1$')
ax.set_ylabel('$f_2$')
ax.set_zlabel('$f_3$')
ax.legend()
fig.tight_layout()