Example: A Real-Valued Multi-Objective Optimization Problem¶

Below is a simple two-variable multi-objective problem to illustrate real-valued optimization with pymoors. We have two continuous decision variables, $x_1$ and $x_2$, both within a given range. We define two objective functions to be minimized simultaneously, and we solve this using the popular NSGA2 algorithm.

Mathematical Formulation¶

Let $\mathbf{x} = (x_1, x_2)$ be our decision variables, each constrained to the interval $[-2, 2]$. We define the following objectives:

$$ \begin{aligned} \min_{\mathbf{x}} \quad &f_1(x_1, x_2) = x_1^2 + x_2^2 \\ \min_{\mathbf{x}} \quad &f_2(x_1, x_2) = (x_1 - 1)^2 + x_2^2 \\ \text{subject to} \quad & -2 \le x_1 \le 2, \\ & -2 \le x_2 \le 2. \end{aligned} $$

Interpretation

1. $f_1$ measures the distance of $\mathbf{x}$ from the origin $(0,0)$ in the 2D plane.
2. $f_2$ measures the distance of (\mathbf{x}) from the point $(1,0)$.

Thus, $\mathbf{x}$ must compromise between being close to $(0,0)$ and being close to $(1,0)$. There is no single point in $[-2,2]^2$ that simultaneously minimizes both distances perfectly (other than at the boundary of these trade-offs), so we end up with a Pareto front rather than a single best solution.

In [2]:

import numpy as np
import matplotlib.pyplot as plt

from pymoors import (
    Nsga2,
    RandomSamplingFloat,
    GaussianMutation,
    SimulatedBinaryCrossover,
    CloseDuplicatesCleaner
)
from pymoors.typing import TwoDArray


def fitness(genes: TwoDArray) -> TwoDArray:
    x1 = genes[:, 0]
    x2 = genes[:, 1]
    # Objective 1: Distance to (0,0)
    f1 = x1**2 + x2**2
    # Objective 2: Distance to (1,0)
    f2 = (x1 - 1)**2 + x2**2
    # Combine the two objectives into a single array
    return np.column_stack([f1, f2])



algorithm = Nsga2(
    sampler=RandomSamplingFloat(min = -2, max = 2),
    crossover=SimulatedBinaryCrossover(distribution_index = 15),
    mutation=GaussianMutation(gene_mutation_rate=0.1, sigma=0.01),
    fitness_fn=fitness,
    num_objectives=2,
    num_constraints=0,
    duplicates_cleaner=CloseDuplicatesCleaner(epsilon=1e-16),
    num_vars=2,
    population_size=200,
    num_offsprings=200,
    num_iterations=100,
    mutation_rate=0.2,
    crossover_rate=0.9,
    keep_infeasible=False,
    lower_bound=-2,
    upper_bound=2,
    verbose = False
)

algorithm.run()
population = algorithm.population

# Plot the results
t = np.linspace(0.0, 1.0, 200)
f1_theo = t**2
f2_theo = (t - 1.0)**2


plt.figure(figsize=(8, 6))
plt.scatter(f1_theo, f2_theo, marker='D', s=18, label='Theoretical Pareto Front')
plt.scatter(population.fitness[:, 0], population.fitness[:, 1], c='r', s=8, marker='o', label='Obtained Front')
plt.xlabel('$f_1$')
plt.ylabel('$f_2$')
plt.title('Two-Target Distance Problem · Pareto Front')
plt.grid(True)
plt.legend()
plt.show()

import numpy as np import matplotlib.pyplot as plt from pymoors import ( Nsga2, RandomSamplingFloat, GaussianMutation, SimulatedBinaryCrossover, CloseDuplicatesCleaner ) from pymoors.typing import TwoDArray def fitness(genes: TwoDArray) -> TwoDArray: x1 = genes[:, 0] x2 = genes[:, 1] # Objective 1: Distance to (0,0) f1 = x1**2 + x2**2 # Objective 2: Distance to (1,0) f2 = (x1 - 1)**2 + x2**2 # Combine the two objectives into a single array return np.column_stack([f1, f2]) algorithm = Nsga2( sampler=RandomSamplingFloat(min = -2, max = 2), crossover=SimulatedBinaryCrossover(distribution_index = 15), mutation=GaussianMutation(gene_mutation_rate=0.1, sigma=0.01), fitness_fn=fitness, num_objectives=2, num_constraints=0, duplicates_cleaner=CloseDuplicatesCleaner(epsilon=1e-16), num_vars=2, population_size=200, num_offsprings=200, num_iterations=100, mutation_rate=0.2, crossover_rate=0.9, keep_infeasible=False, lower_bound=-2, upper_bound=2, verbose = False ) algorithm.run() population = algorithm.population # Plot the results t = np.linspace(0.0, 1.0, 200) f1_theo = t**2 f2_theo = (t - 1.0)**2 plt.figure(figsize=(8, 6)) plt.scatter(f1_theo, f2_theo, marker='D', s=18, label='Theoretical Pareto Front') plt.scatter(population.fitness[:, 0], population.fitness[:, 1], c='r', s=8, marker='o', label='Obtained Front') plt.xlabel('$f_1$') plt.ylabel('$f_2$') plt.title('Two-Target Distance Problem · Pareto Front') plt.grid(True) plt.legend() plt.show()