遗传算法求三元函数极值

遗传算法是一种基于自然选择和遗传学原理的优化算法，可以用于求解函数极值问题。下面是遗传算法求解三元函数极值的步骤：

1.定义适应度函数：对于三元函数f(x1,x2,x3)=x1*x1-x1*x2+x3，我们可以将其作为适应度函数，即fitness(x1,x2,x3)=f(x1,x2,x3)。

2.初始化种群：随机生成一定数量的个体，每个个体由三个实数编码表示，即x1、x2、x3。

3.选择操作：采用轮盘赌选择算子，根据个体适应度大小选择优秀的个体。

4.交叉操作：采用单点交叉算子，对选出的个体进行交叉操作，生成新的个体。

5.变异操作：对新生成的个体进行变异操作，增加种群的多样性。

6.评估函数：计算每个个体的适应度值。

7.选择优秀个体：根据适应度值选择优秀的个体。

8.重复执行步骤3-7，直到达到预设的迭代次数或找到最优解。

下面是Python代码实现：

import random

# 定义适应度函数
def fitness(x1, x2, x3):
    return x1 * x1 - x1 * x2 + x3

# 初始化种群
def init_population(pop_size):
    population = []
    for i in range(pop_size):
        x1 = random.uniform(-10, 10)
        x2 = random.uniform(-10, 10)
        x3 = random.uniform(-10, 10)
        population.append((x1, x2, x3))
    return population

# 选择操作
def selection(population):
    fitness_list = [fitness(*individual) for individual in population]
    total_fitness = sum(fitness_list)
    probability_list = [fitness / total_fitness for fitness in fitness_list]
    selected_population = []
    for i in range(len(population)):
        selected_individual = random.choices(population, weights=probability_list)[0]
        selected_population.append(selected_individual)
    return selected_population

# 交叉操作
def crossover(population):
    offspring_population = []
    for i in range(len(population)):
        parent1 = population[i]
        parent2 = random.choice(population)
        crossover_point = random.randint(0, 2)
        offspring = parent1[:crossover_point] + parent2[crossover_point:]
        offspring_population.append(offspring)
    return offspring_population

# 变异操作
def mutation(population, mutation_rate):
    mutated_population = []
    for individual in population:
        if random.random() < mutation_rate:
            mutated_individual = list(individual)
            mutated_individual[random.randint(0, 2)] += random.uniform(-1, 1)
            mutated_population.append(tuple(mutated_individual))
        else:
            mutated_population.append(individual)
    return mutated_population

# 遗传算法求解三元函数极值
def genetic_algorithm(pop_size, max_iter, mutation_rate):
    population = init_population(pop_size)
    for i in range(max_iter):
        selected_population = selection(population)
        offspring_population = crossover(selected_population)
        mutated_population = mutation(offspring_population, mutation_rate)
        population = mutated_population
    best_individual = max(population, key=lambda x: fitness(*x))
    return best_individual

# 测试
best_individual = genetic_algorithm(100, 1000, 0.1)
print("最优解：", best_individual)
print("最优解对应的函数值：", fitness(*best_individual))