目录

• 1 问题定义
• 2 算法步骤
• 3 代码
• 4 效果

1 问题定义

本博客以最小化问题为例
f1=x2f2=(x−2)2min⁡f=(f1(x),f2(x))\begin{aligned} f_1 & = x ^2 \\ f_2 & = (x - 2) ^2 \\ \min f & = (f_1(x), f_2(x)) \end{aligned} f1​f2​minf​=x2=(x−2)2=(f1​(x),f2​(x))​

代码

def func1(population):res = population[:, 0] ** 2return resdef func2(population):res = (population[:, 0] - 2) ** 2return res

2 算法步骤

1. EPEPEP：外部种群，大小一般为10∗size10 * size10∗size(sizesizesize是种群大小)
2. λ\lambdaλ：权重向量，本文所提到的权重向量需要均匀分布，可以用numpy正态分布产生。
3. zzz：是当代种群，每个目标的最小值
4. B(i)B(i)B(i)：当前点的邻居索引，邻居的计算使用欧氏距离，可以调用sklearn中的库完成
5. gte(x∣λj,z∗)=max⁡{λij∣fi(x)−z∗∣}g^{te}(x|\lambda ^j, z^*) = \max\{\lambda_i^j|f_i(x) - z^*|\}gte(x∣λj,z∗)=max{λij​∣fi​(x)−z∗∣}，其中1≤i≤m1 \leq i \leq m1≤i≤m（切比雪夫）

因为本博客以最小化问题为例，而原论文则是求解最大化问题，因此上方图片中要做如下修改
Step 2.3中的 zj>fj(y′)z_j > f_j(y')zj​>fj​(y′)

注意：当目标数m<=2时适合用切比雪夫

3 代码

import numpy as np
import pandas as pd
from geatpy.core.ndsortESS import ndsortESS  # 非支配排序
from sklearn.neighbors import NearestNeighbors   # K近邻分类器
from geatpy.operators.mutation.Mutpolyn import Mutpolyn  # 多项式变异
from geatpy.operators.recombination.Recsbx import Recsbx  # 模拟二进制交叉
from geatpy.visualization.PointScatter import PointScatter  # 画图
from untils.SCH import func1, func2# 通过Euclidean distance产生近邻
def get_neighbor(matrix, n_neighbors = 2):nbrs = NearestNeighbors(n_neighbors=n_neighbors, algorithm='kd_tree', metric='euclidean').fit(matrix)distances, indices = nbrs.kneighbors(matrix)return indices   # 返回邻居索引# 在邻居集合里挑选两个邻居, 返回邻居索引
def select_neighbor(neighbor_indexs, prob_neighbor = 0.9):indices = []for index in neighbor_indexs:if len(indices) > 2: breakprobability = np.random.rand()if probability < prob_neighbor:indices.append(index)if len(indices) == 0:return neighbor_indexs[0:2]return indices   # 返回邻居索引# 切比雪夫方法更新邻居解
def update_neighbor(weight_vector, neighbor_func, z, y_func):# 第一行 对应的为y的相关值copy_func = np.copy(neighbor_func)copy_func = np.abs(copy_func - z)     # 函数值 - 参考点te = np.multiply(weight_vector, copy_func)    # 叉乘y_te = np.multiply(weight_vector, np.abs(y_func-z))row_max = np.max(te, axis=1)bad_neighbor = []n = row_max.shape[0]for i in range(n):if np.max(y_te) <= row_max[i]:   # 需要对邻居更新bad_neighbor.append(i)return bad_neighbordef update_external_population(pop, func, y, y_func):  # pop: 外部种群；func: 外部种群函数值if pop is None:pop = yfunc = y_funcelse:pop = np.row_stack((pop, y))func = np.row_stack((func, y_func))levels, criLevel = ndsortESS(func)levels = levels.astype('i4')  # float -> intpd_levels = pd.DataFrame(levels)  # numpy -> pandasrank_group = pd_levels.groupby(0).groups  # 等级分组best_rank = rank_group[1].values  # 最好等级的所有下标pop = pop[best_rank, :]func = func[best_rank, :]if func.shape[0] > K * size:r = np.random.randint(0, K * size, 1)pop = np.delete(pop, r, axis=0)func = np.delete(func, r, axis=0)return pop, funcif __name__ == '__main__':size = 50  # 种群大小dim = 2     # 目标函数的个数n = 1      # 决策变量的个数n_neighbor = 50    # 权重向量近邻的个数lb = np.full(n, -10)   # 自变量下限ub = np.full(n, 10)    # 自变量上限varTypes = np.full(n, 0)    # 离散 or 连续iteration = 250   # 迭代次数K = 2    # 外部种群大小 = K * sizenp.random.seed(1)# ===================== 产生均匀分布的权重向量 =======================weight_vector = np.random.randn(size, dim)ss = np.sum(weight_vector, axis=1)for k in range(dim):weight_vector[:, k] = weight_vector[:, k] / ss# ===================== 产生均匀分布的权重向量 =======================all_neighbor = get_neighbor(weight_vector, n_neighbor)# ========================= 产生随机种群 ===========================x = np.random.uniform(lb, ub, (size, n))func = np.column_stack((func1(x), func2(x)))ideal_points = np.min(func, axis=0)  # 所有函数值的最小值,即理想点# ========================= 产生随机种群 ===========================exter_pop = None   # 外部种群exter_obj = None   # 外部种群函数值PS = PointScatter(dim, True, True, "MOEA/D", ['F1', 'F2', 'F3'], saveName=None)for i in range(iteration):print(i)for s in range(size):# =========================  挑选两个邻居  ==========================neighbor = all_neighbor[s, :]   # 当前个体的邻居索引two_neighbor = np.random.choice(neighbor, size = 2, replace=False)neighborsX = x[two_neighbor, :]# =========================  挑选两个邻居  ==========================# ========================  进行交叉变异操作  =========================recsbx = Recsbx(XOVR=1, Half_N=True, n=20, Parallel=True)  # 交叉y = recsbx.do(neighborsX)mutation = Mutpolyn(Pm=1/n, DisI=20)  # 变异y = mutation.do(Encoding='RI', OldChrom=y, FieldDR=np.array([lb, ub, varTypes]))# ========================  进行交叉变异操作  =========================# =============================  更新z  ==============================y_func = np.column_stack((func1(y), func2(y)))for d in range(dim):ideal_points[d] = np.minimum(ideal_points[d], y_func[0, d])# =============================  更新z  ==============================# ===========================  更新邻居解  ============================weight_vector_neighbor = weight_vector[neighbor, :]      # 邻居的权重向量neighborsFunc = func[neighbor, :]          # 邻居的函数矩阵bad_neighbor = update_neighbor(weight_vector_neighbor, neighborsFunc,ideal_points, y_func)x[bad_neighbor, :] = yfunc[bad_neighbor, :] = y_func# ===========================  更新邻居解  ============================# ===========================  更新外部种群  ============================exter_pop, exter_obj = update_external_population(exter_pop, exter_obj, y, y_func)# ===========================  更新外部种群  ============================right_label = 'size: ' + str(K * size) + '\n' + 'iteration: ' + str(iteration)PS.add(exter_obj, color='purple', label=right_label)PS.draw()

4 效果