Differential Evolution Algorithm: Python Implementation and Detailed Explanation

import numpy as npimport matplotlib.pyplot as pltclass DifferentialEvolution: “”” Differential Evolution Algorithm Implementation Class Parameters: objective_func: Objective function bounds: List of boundaries for each variable, e.g.,[(min1, max1), (min2, max2), …] pop_size: Population size(default 20) F: Mutation factor(default 0.8) CR: Crossover probability(default 0.7) max_iter: Maximum number of iterations(default 1000) verbose: Whether to print iteration information(default False) “”” def __init__(self, objective_func, bounds, pop_size=20, F=0.8, CR=0.7, max_iter=1000, verbose=False): self.objective_func = objective_func self.bounds = np.array(bounds) self.pop_size = pop_size self.F = F self.CR = CR self.max_iter = max_iter self.verbose = verbose self.dim = len(bounds) # Problem dimension self.min_b, self.max_b = np.asarray(bounds).T # Extract boundaries self.diff = np.fabs(self.min_b – self.max_b) # Calculate boundary difference # Initialize population and fitness self.population = None self.fitness = None self.best_solution = None self.best_fitness = float(‘inf’) self.history = [] # Record optimal fitness history def initialize_population(self): “””Initialize population“”” # Randomly generate the initial population within the range [0,1] self.population = np.random.rand(self.pop_size, self.dim) # Scale the population to the actual boundary range self.population = self.min_b + self.population * self.diff # Calculate initial fitness self.fitness = np.array([self.objective_func(ind) for ind in self.population]) # Record the best individual best_idx = np.argmin(self.fitness) self.best_solution = self.population[best_idx] self.best_fitness = self.fitness[best_idx] self.history.append(self.best_fitness) def mutate(self): “””Mutation operation(UsingDE/rand/1strategy)””” mutants = np.zeros_like(self.population) for i in range(self.pop_size): # Randomly select three individuals different from the current individual idxs = [idx for idx in range(self.pop_size) if idx != i] a, b, c = self.population[np.random.choice(idxs, 3, replace=False)] # Generate mutated individuals(Mutation operation) mutants[i] = a + self.F * (b – c) # Boundary handling: Ensure that the mutated individuals are within the boundary range mutants[i] = np.clip(mutants[i], self.min_b, self.max_b) return mutants def crossover(self, mutants): “””Crossover operation(Binomial crossover)””” trials = np.zeros_like(self.population) for i in range(self.pop_size): # Randomly select at least one dimension for crossover cross_points = np.random.rand(self.dim) < self.CR if not np.any(cross_points): cross_points[np.random.randint(0, self.dim)] = True # Perform crossover trials[i] = np.where(cross_points, mutants[i], self.population[i]) return trials def select(self, trials): “””Select operation“”” trial_fitness = np.array([self.objective_func(ind) for ind in trials]) # Compare the fitness of trial individuals and original individuals improved = trial_fitness < self.fitness self.population[improved] = trials[improved] self.fitness[improved] = trial_fitness[improved] # Update global optimal solution current_best_idx = np.argmin(self.fitness) if self.fitness[current_best_idx] < self.best_fitness: self.best_solution = self.population[current_best_idx] self.best_fitness = self.fitness[current_best_idx] self.history.append(self.best_fitness) def evolve(self): “””Execute evolution process“”” self.initialize_population() for iter in range(self.max_iter): mutants = self.mutate() # Mutation trials = self.crossover(mutants) # Cross self.select(trials) # Select if self.verbose and (iter % 100 == 0 or iter == self.max_iter – 1): print(f”Iteration {iter: Best Fitness = {self.best_fitness:.6f}”) return self.best_solution, self.best_fitness def plot_convergence(self): “””Plot convergence curve“”” plt.figure(figsize=(10, 6)) plt.plot(self.history, ‘b-‘, linewidth=2) plt.title(‘Convergence Curve’, fontsize=14) plt.xlabel(‘Iteration’, fontsize=12) plt.ylabel(‘Best Fitness’, fontsize=12) plt.grid(True) plt.show()# Example: Test the Differential Evolution Algorithmif __name__ == “__main__”: # Define the objective function(Here we use theRastriginfunction) def rastrigin(x): “””Rastriginfunction: Multimodal function, commonly used to test optimization algorithms“”” A = 10 return A * len(x) + sum([(xi ** 2 – A * np.cos(2 * np.pi * xi)) for xi in x]) # Define variable boundaries(5dimensional problem) bounds = [(-5.12, 5.12)] * 5 # Create an instance of Differential Evolution de = DifferentialEvolution(rastrigin, bounds, pop_size=50, F=0.8, CR=0.9, max_iter=1000, verbose=True) # Execute optimization best_solution, best_fitness = de.evolve() # Output results print(“\nOptimization Results:”) print(f”Best Solution: {best_solution}”) print(f”Best Fitness: {best_fitness:.6f}”) # Plot convergence curve de.plot_convergence()