### 使用Python实现NSGA-II算法进行三目标函数优化 为了满足对三个目标函数的多目标优化需求，可以利用面向对象编程方法构建NSGA-II框架。下面展示的是一个简化版的NSGA-II类结构以及具体的应用实例。 #### NSGA-II 类定义 ```python import numpy as np from scipy.optimize import differential_evolution class Individual: """个体表示""" def __init__(self, genes): self.genes = genes # 基因编码 self.objectives = None # 目标值 def evaluate(self, objectives_func): """评估个体适应度""" self.objectives = objectives_func(self.genes) class Population: """种群管理""" def __init__(self, size=100, dimensions=30, lower_bound=-5.0, upper_bound=5.0): self.size = size self.individuals = [] for _ in range(size): individual = Individual(np.random.uniform(lower_bound, upper_bound, (dimensions,))) self.individuals.append(individual) def non_dominated_sorting(population): """执行非支配排序""" fronts = [[]] S = {} n = {} rank = {} for p in population.individuals: S[p] = [] # 被p支配的集合 n[p] = 0 # 支配p的数量 for q in population.individuals: if all(p.objectives[i] <= q.objectives[i] for i in range(len(q.objectives))): if not any(p.objectives[j] > q.objectives[j] for j in range(len(q.objectives))): S[p].append(q) elif all(q.objectives[k] <= p.objectives[k] for k in range(len(p.objectives))): if not any(q.objectives[l] > p.objectives[l] for l in range(len(p.objectives))): n[p] += 1 if n[p] == 0: rank[p] = 0 fronts[0].append(p) i = 0 while fronts[i]: Q = [] for p in fronts[i]: for q in S[p]: n[q] -= 1 if n[q] == 0: rank[q] = i + 1 Q.append(q) i += 1 fronts.append(Q) del fronts[-1] return fronts def crowding_distance_assignment(front): """拥挤距离计算""" distance = {member: 0 for member in front} obj_num = len(front[0].objectives) for o in range(obj_num): sorted_front = sorted(front, key=lambda x: x.objectives[o]) distance[sorted_front[0]] = float('inf') distance[sorted_front[-1]] = float('inf') max_obj_val = max([individ.objectives[o] for individ in front]) min_obj_val = min([individ.objectives[o] for individ in front]) for i in range(1, len(sorted_front)-1): dist = abs((sorted_front[i+1].objectives[o]-sorted_front[i-1].objectives[o])/(max_obj_val-min_obj_val)) distance[sorted_front[i]] += dist return distance def select_parents(population, mating_pool_size): """选择父代""" selected_individuals = [] fronts = non_dominated_sorting(population) remaining spots = mating_pool_size for f_idx, front in enumerate(fronts): if remaining_spots >= len(front): selected_individuals.extend(front) remaining_spots -= len(front) else: distances = crowding_distance_assignment(front) ordered_by_crowding_dist = sorted(distances.items(), key=lambda item:item[1], reverse=True)[:remaining_spots] selected_individuals.extend([item[0] for item in ordered_by_crowding_dist]) break return selected_individuals[:mating_pool_size] def crossover(parent_1_genes, parent_2_genes, prob_cross=.9): """交叉操作""" offspring_genes = [] for gene_p1, gene_p2 in zip(parent_1_genes, parent_2_genes): if np.random.rand() < prob_cross: alpha = np.random.rand() child_gene = alpha * gene_p1 + (1-alpha)*gene_p2 else: child_gene = gene_p1 if np.random.randint(2)==0 else gene_p2 offspring_genes.append(child_gene) return offspring_genes def mutate(gene_vector, mutation_prob=.1, scale_factor=.2): """变异操作""" mutated_genes = [ g + ((np.random.randn()*scale_factor) if np.random.rand()<mutation_prob else 0.) for g in gene_vector ] return mutated_genes class NsgaII: """NSGA-II 主体逻辑封装""" def __init__(self, objective_functions, pop_size=100, generations=250, cross_rate=.9, mut_rate=.1, dim=30, lb=-5., ub=5.): self.pop_size = pop_size self.generations = generations self.cross_rate = cross_rate self.mut_rate = mut_rate self.population = Population(pop_size, dim, lb, ub) self.objective_funcs = objective_functions def run(self): """运行进化过程""" for gen in range(self.generations): # 计算当前群体内所有成员的目标向量 for indv in self.population.individuals: indv.evaluate(self.objective_funcs) parents = select_parents(self.population, int(self.pop_size/2)) new_population = [] for idx in range(int(self.pop_size / 2)): parent_a, parent_b = np.random.choice(parents, 2, replace=False) children_genes = crossover(parent_a.genes, parent_b.genes, self.cross_rate) children_genes_mutated = mutate(children_genes, self.mut_rate) child_indivdual = Individual(children_genes_mutated) new_population.append(child_indivdual) combined_pop = Population( individuals=self.population.individuals + new_population ) next_gen_members = select_parents(combined_pop, self.pop_size) self.population.individuals = list(next_gen_members) final_evaluated = [i.evaluate(self.objective_funcs) for i in self.population.