python 优化算法

以下是几种常见的Python优化算法的示例：

1. 梯度下降法（Gradient Descent）

def gradient_descent(f, df, x0, learning_rate, num_iterations):
    x = x0
    for i in range(num_iterations):
        gradient = df(x)
        x -= learning_rate * gradient
    return x

# 示例：求解函数 f(x) = x^2 的最小值
f = lambda x: x**2
df = lambda x: 2*x
x0 = 1.0
learning_rate = 0.1
num_iterations = 100
result = gradient_descent(f, df, x0, learning_rate, num_iterations)
print("最小值：", result)

2. 共轭梯度法（Conjugate Gradient）

import numpy as np

def conjugate_gradient(A, b, x0, num_iterations):
    x = x0
    r = b - np.dot(A, x)
    p = r
    for i in range(num_iterations):
        Ap = np.dot(A, p)
        alpha = np.dot(r, r) / np.dot(p, Ap)
        x += alpha * p
        r_new = r - alpha * Ap
        beta = np.dot(r_new, r_new) / np.dot(r, r)
        p = r_new + beta * p
        r = r_new
    return x

# 示例：求解线性方程组 Ax = b
A = np.array([[3, 2], [2, 6]])
b = np.array([2, -8])
x0 = np.array([0, 0])
num_iterations = 10
result = conjugate_gradient(A, b, x0, num_iterations)
print("解：", result)

3. 遗传算法（Genetic Algorithm）

import random

def genetic_algorithm(fitness_func, num_generations, population_size, mutation_rate):
    population = [random.randint(0, 1) for _ in range(population_size)]
    for _ in range(num_generations):
        fitness_scores = [fitness_func(individual) for individual in population]
        new_population = []
        for _ in range(population_size):
            parent1, parent2 = random.choices(population, weights=fitness_scores, k=2)
            child = crossover(parent1, parent2)
            child = mutate(child, mutation_rate)
            new_population.append(child)
        population = new_population
    best_individual = max(population, key=fitness_func)
    return best_individual

# 示例：求解二进制字符串中1的个数最大的个体
def fitness_func(individual):
    return sum(individual)

def crossover(parent1, parent2):
    crossover_point = random.randint(1, len(parent1) - 1)
    child = parent1[:crossover_point] + parent2[crossover_point:]
    return child

def mutate(individual, mutation_rate):
    mutated_individual = []
    for gene in individual:
        if random.random() < mutation_rate:
            mutated_individual.append(1 - gene)
        else:
            mutated_individual.append(gene)
    return mutated_individual

num_generations = 100
population_size = 100
mutation_rate = 0.01
result = genetic_algorithm(fitness_func, num_generations, population_size, mutation_rate)
print("最优个体：", result)