Decomposition-Based Multi-Objective Optimization (MOEA/D)

This guide demonstrates how to solve multi-objective optimization problems by decomposing them into a set of single-objective subproblems using MOEAD and scalarization functions such as PBI.

Example Code

import numpy as np

from opytimizer import Opytimizer
from opytimizer.core import Function
from opytimizer.core.stopping import MaxIterations
from opytimizer.math.aggregation import PBI
from opytimizer.optimizers.multi_objective.evolutionary import MOEAD
from opytimizer.spaces import SearchSpace
from opytimizer.utils.reference_vectors import das_dennis


def zdt3(x: np.ndarray) -> np.ndarray:
    """
    ZDT3 benchmark problem
    References:
    Zhang, Q., & Li, H. (2007). MOEA/D: A multiobjective evolutionary algorithm based on decomposition.
    IEEE Transactions on evolutionary computation, 11(6), 712-731.
    """

    # x has (n, 1) shape, where n is the number of decision variables
    x = x.flatten()
    f1 = x[0]
    n = x.shape[0]
    g = 1 + (9 * np.sum(x[1:])) / (n - 1)
    f2 = g * (1 - np.sqrt(f1 / g) - (f1 / g) * np.sin(10 * np.pi * x[0]))

    return [f1, f2]


# Random seed for experimental consistency
np.random.seed(0)

n_variables = 30
n_objectives = 2
weights, n_agents = das_dennis(2, 99)

# Lower and upper bounds (has to be the same size as `n_variables`)
lower_bound = [0] * n_variables
upper_bound = [1] * n_variables

pbi = PBI()

# Creates the space, optimizer and function
space = SearchSpace(n_agents, n_variables, n_objectives, lower_bound, upper_bound)
optimizer = MOEAD(weight_vectors=weights, decomposition_method=pbi)
function = Function(zdt3)

# Bundles every piece into Opytimizer class
opt = Opytimizer(space, optimizer, function, save_agents=False)

# Runs the optimization task
opt.start(MaxIterations(250))

Example Code​

Example Code