MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

1 问题定义

本博客以最小化问题为例
$$\begin{array}{rl}{f}_1& =x^2\\ f_2& =(x-2{)}^2\\ \min f& =(f_1(x),f_2(x))\end{array}$$

代码

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


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

2 算法步骤

1. $EP$：外部种群，大小一般为$10\ast size$($size$是种群大小)
2. $\lambda$：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. $z$：是当代种群，每个目标的最小值
4. $B(i)$：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. $g^{te}(x|{\lambda }^j,z^{\ast })=\max \{{\lambda }_i^j|f_i(x)-z^{\ast }|\}$，其中$1\le i\le m$（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 $z_j>f_j(y^{\prime })$

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

# 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_neighbor

def 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 = y
        func = y_func
    else:
        pop = np.row_stack((pop, y))
        func = np.row_stack((func, y_func))

    levels, criLevel = ndsortESS(func)
    levels = levels.astype('i4')  # float -> int
    pd_levels = pd.DataFrame(levels)  # numpy -> pandas
    rank_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, func

if __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 * size
    np.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, :] = y
            func[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