Differential Evolution in Evolutionary Algorithms (Differential Evolution)

Table of contents

introduction

Differential Evolution (DE) is a global optimization algorithm that can be used to solve complex optimization problems. It is derived from genetic algorithms and evolutionary strategies, and searches for optimal solutions by simulating the evolutionary process in nature. The differential evolution algorithm is widely used in function optimization, parameter optimization, machine learning and other fields, and has good robustness and global search capabilities.

Algorithm principle

The differential evolution algorithm searches and optimizes based on the differences between individuals. It generates new solutions through differential mutation of candidate solutions and uses a fitness function to evaluate the quality of the solution. The following are the basic steps of the differential evolution algorithm:

Initialization population: Randomly generate a population of initial candidate solutions.
Mutation operation: For each individual, select three different individuals as mutation vectors, and adjust the magnitude of the mutation vector according to the mutation factor.
Crossover operation: Cross the mutation vector with the current individual to generate a new solution.
Selection operation: Compare the quality of the new solution and the current individual according to the fitness function, and select the better solution as the next generation individual.
Termination condition: The algorithm is terminated when the predetermined number of iterations is reached or the stopping criterion is met, and the optimal solution is returned.

Algorithm characteristics

The differential evolution algorithm has the following characteristics:

Simple and effective: The differential evolution algorithm does not depend on the specific nature of the problem and is suitable for various optimization problems.
Global search: The differential evolution algorithm has good global search capabilities and can find the global optimal solution to the problem.
Robustness: The differential evolution algorithm is relatively insensitive to the selection of initial solutions and parameter settings, and has good robustness.
Low memory consumption: The differential evolution algorithm only needs to store information about the current individual and new solutions, and has low memory consumption.

The following is an example code that uses Python to implement the differential evolution algorithm:

pythonCopy codeimport random
import numpy as np
def differential_evolution(fitness_func, bounds, population_size=50, max_generations=100, crossover_rate=0.7, differential_weight=0.5):
    # 初始化种群
    population = np.random.uniform(bounds[0], bounds[1], (population_size, len(bounds)))
    for generation in range(max_generations):
        for i in range(population_size):
            # 选择三个不同的个体作为变异向量
            candidates = [j for j in range(population_size) if j != i]
            a, b, c = random.sample(candidates, 3)
            # 变异操作
            mutant = population[a] + differential_weight * (population[b] - population[c])
            # 交叉操作
            trial = np.copy(population[i])
            for j in range(len(bounds)):
                if random.random() < crossover_rate:
                    trial[j] = mutant[j]
            # 选择操作
            if fitness_func(trial) < fitness_func(population[i]):
                population[i] = trial
    # 返回最优解
    best_solution = population[np.argmin([fitness_func(x) for x in population])]
    best_fitness = fitness_func(best_solution)
    return best_solution, best_fitness
# 示例函数：Rastrigin函数
def rastrigin(x):
    n = len(x)
    return 10 * n + sum([(xi ** 2 - 10 * np.cos(2 * np.pi * xi)) for xi in x])
# 设置函数的边界和参数
bounds = [(-5.12, 5.12)] * 10
population_size = 100
max_generations = 200
crossover_rate = 0.9
differential_weight = 0.5
# 执行差分进化算法
best_solution, best_fitness = differential_evolution(rastrigin, bounds, population_size, max_generations, crossover_rate, differential_weight)
# 输出结果
print("最优解:", best_solution)
print("最优适应度:", best_fitness)

Please note that the above code is a simplified example and is only used to demonstrate the basic principles and steps of the differential evolution algorithm. In actual use, it may be necessary to adjust parameters and optimize the algorithm according to specific problems.

Applications

Differential evolution algorithms have been successfully used in many fields. The following are some typical application cases:

Function optimization: The differential evolution algorithm performs well in mathematical function optimization and can find the global optimal solution to complex functions.
Parameter optimization: Differential evolution algorithm is widely used for parameter optimization in machine learning and deep learning, such as weight optimization of neural networks.
Feature selection: Differential evolution algorithm can be used for feature selection to select the optimal feature subset from a large number of features for pattern recognition and data mining tasks.
Machine parameter tuning: Differential evolution algorithms are also commonly used for hyperparameter tuning of machine learning models to improve model performance and generalization capabilities.

The following is an example code for feature selection using the differential evolution algorithm:

pythonCopy codeimport random
import numpy as np
def differential_evolution_feature_selection(population, fitness_func, bounds, max_generations=100, crossover_rate=0.7, differential_weight=0.5):
    # 初始化种群
    population_size, num_features = population.shape
    for generation in range(max_generations):
        for i in range(population_size):
            # 选择三个不同的个体作为变异向量
            candidates = [j for j in range(population_size) if j != i]
            a, b, c = random.sample(candidates, 3)
            # 变异操作
            mutant = population[a] + differential_weight * (population[b] - population[c])
            # 交叉操作
            trial = np.copy(population[i])
            for j in range(num_features):
                if random.random() < crossover_rate:
                    trial[j] = mutant[j]
            # 选择操作
            if fitness_func(trial) < fitness_func(population[i]):
                population[i] = trial
    # 返回最优解
    best_solution = population[np.argmin([fitness_func(x) for x in population])]
    best_fitness = fitness_func(best_solution)
    return best_solution, best_fitness
# 示例函数：简单的特征选择问题
def feature_selection(x):
    # 假设适应度函数为特征的数量
    return sum(x)
# 设置参数
population_size = 50
num_features = 10
max_generations = 100
crossover_rate = 0.7
differential_weight = 0.5
bounds = [(0, 1)] * num_features
# 初始化种群
population = np.random.randint(2, size=(population_size, num_features))
# 执行差分进化算法进行特征选择
best_solution, best_fitness = differential_evolution_feature_selection(population, feature_selection, bounds, max_generations, crossover_rate, differential_weight)
# 输出结果
print("最优解:", best_solution)
print("最优适应度:", best_fitness)

The above code assumes that the value of the feature is 0 or 1, and the fitness function is the number of selected features. Through the iterative process of the differential evolution algorithm, the final optimal solution is the selected feature. Please note that the above code is for reference only. In actual use, the fitness function needs to be defined and parameters adjusted according to specific problems.

Summarize

The differential evolution algorithm is a powerful global optimization algorithm that can be applied to various optimization problems. It continuously generates and selects new solutions by simulating the evolutionary process to find the optimal solution to the problem. The differential evolution algorithm has the characteristics of simplicity, effectiveness, global search, robustness, and low memory consumption. It has achieved good application results in fields such as function optimization, parameter optimization, and feature selection. In the future, with the continuous improvement and expansion of the algorithm, the differential evolution algorithm will exert its advantages in more fields and provide more possibilities for solving complex optimization problems.