作為一名非數學、非計算機專業的野生研究僧程序員，在學習和實踐多目標優化時，遇到了各種困難，加之相關方向的交流資源有限，也使得整個過程顯得緩慢。在此，非常感謝西電曉風（https://blog.csdn.net/qq_40434430/article/details/82876572）及其他各路貢獻知識的大神給予的啟發、交流，他的嚴謹、謙虛、熱情讓我能不拋棄不放棄，入了門。

為何要專門講解NSGA2非支配排序函數

多目標優化中的經典是NSGA2，NSGA2的經典是non-dominated sort。非支配排序做好了，才能談后面的一切（這句話的領悟，非生而知之，而是在各種試錯、奔潰的過程中領略到的）。掌握了NSGA2才能談后面的其他各類算法，如：NSGA3，ε-NSGA2，MOEA，MOEA/D（這里的提法也并非完全正確，他們直接部分并沒有強耦合關系，僅為個人一己之見）。

非支配排序函數的基本原理

首先看一下數學定義：

需要說明幾點潛在內涵：

• 都是求最小化問題（一個約定俗成的規則）；
• 所有的 $f_i(X_a)$ 小于或者等于$f_i(X_b)$ ，且至少存在一個 $f_i(X_a)$ 完全小于$f_i(X_b)$
• $f_i(X_a)$ 完全等于$f_i(X_b)$時，$X_a$和$X_b$不存在支配關系（劃重點：這個在實際的進化過程中，會出現很多，目前網上很多的代碼忽略了這一點，會導致在已有Deb大神原始NSGA2測試問題上運行沒問題，但是轉換到現實問題中會出現很多的問題，如：不收斂、不穩定等。為什么會出現這樣的問題呢？因為Deb大神采用的是標準遺傳算子中的一種，即：實數編碼和多項式交叉，且論文中定義的PC很大，世代之間出現完全復制的可能性小。然而，現實中應用改造問題，很多時候需要改造掉遺傳算子，即：非標遺傳算子，從而就會展現出各種奇葩問題。之所以在這里強調這問題，是因為我自己在這里耗費了很多的精力，各種調試，找不出原因，最終發現是因為一個等于號坑了我。也特別感謝課題組Jing Huang給予的耐心調試和討論。）

如何測試非支配排序函數

我們需要構建專門針對非支配排序函數的測試類，從而驗證函數的正確性，構建測試類的關鍵是在于輸入的測試評價值Array的有效性。我通過大量的實驗，提煉出基礎的、正確的、報錯的三個測試評價值Array。

• 基礎的測試評價值Array，大家可以用來自行推演，該值參考至鄭金華老師的書（主觀評價：鄭老師的書是目前國內對此塊做了系統介紹的書，他的治學風格嚴謹、務實，而非論文的直接堆砌，建議入門學生可以從這本書開始，論文堆砌的書可以到高階階段再學習）
• 正確的測試評價值Array，跑ZDT3的時候從內存中Debug Copy的。
• 報錯的測試評價值Array，跑ZDT6的時候從內存中Debug Copy的。以下給出的代碼中test_fast_non_dominated_sort_error函數會顯示出他錯在哪。

import random import numpy as np from matplotlib.ticker import MultipleLocator import matplotlib.pyplot as pltclass Test_class():def __init__(self,f_num):self.f_num=f_num# 測試代碼Y1 = [9, 7, 5, 4, 3, 2, 1, 10, 8, 7, 5, 4, 3, 10, 9, 8, 7, 10, 9, 8]Y2 = [1, 2, 4, 5, 6, 7, 9, 1, 5, 6, 7, 8, 9, 5, 6, 7, 9, 6, 7, 9]self.objectives_fitness_zjh=np.array([Y1,Y2]).Tself.objectives_fitness_8 = np.array([[0.6373723585181119, 9.089424920752537],[0.6307563745957109, 9.484134522661321],[0.6307726564027054, 9.370453232401315],[0.9214017573731662, 8.974173267351892],[0.8106208092655269, 9.00814519794432],[0.6308299859236132, 9.381311843663337],[0.9996933421004693, 8.998732212375378],[0.8106208092655269, 9.087298161567794],[0.9968206553777186, 9.020037858133483],[0.9503113437004427, 9.04519027298749],[0.9214017573731662, 8.974173267351892],[0.9214017573731662, 9.00187266065025],[0.6307563745957109, 9.493829448943414],[0.999999999999489, 9.180616877016963],[0.8150090640161689, 9.041622216379706],[0.9990805452551389, 8.980910429010232],[0.9979468094812165, 9.04561922020985],[0.7527617476769539, 9.136335451211739],[0.6364241984468356, 9.20992553291975],[0.8106208092655269, 9.083868693443087]])self.objectives_fitness_20 = np.array([[0.01783689,1.02469787],[0.04471213,0.93037726],[0.0236877,0.96322404],[0.04938809,0.92641409],[0.07031985,0.83355839],[0.05199109,0.88398541],[0.05006115,0.91107188],[0.,1.18579768],[0.05358649,0.85849188],[0.01657041,1.0721905,],[0.10409921,0.84829613],[0.06643826,0.84941993],[0.05358649,0.89876683],[0.01740193,1.09732744],[0.04938809,0.94687415],[0.05199109,0.93536007],[0.02400905,1.11603965],[0.05358649,0.89107165],[0.,1.23996387],[0.01783689,1.11625582]])def test_fast_non_dominated_sort_1(self, objectives_fitness):#對華電小扎扎寫的非支配排序的修改和調整# https: // blog.csdn.net / qq_36449201 / article / details / 81046586fronts = [] # Pareto前沿面fronts.append([])set_sp = []npp = np.zeros(2*10)rank = np.zeros(2*10)for i in range(2 * 10):temp = []for j in range(2 * 10):if j != i:if (objectives_fitness[j][0] >= objectives_fitness[i][0] and objectives_fitness[j][1] > objectives_fitness[i][1]) or \(objectives_fitness[j][0] > objectives_fitness[i][0] and objectives_fitness[j][1] >= objectives_fitness[i][1]) or \(objectives_fitness[j][0] >= objectives_fitness[i][0] and objectives_fitness[j][1] >= objectives_fitness[i][1]):temp.append(j)elif (objectives_fitness[i][0] >= objectives_fitness[j][0] and objectives_fitness[i][1] > objectives_fitness[j][1]) or \(objectives_fitness[i][0] > objectives_fitness[j][0] and objectives_fitness[i][1] >= objectives_fitness[j][1]) or \(objectives_fitness[j][0] > objectives_fitness[i][0] and objectives_fitness[j][1] > objectives_fitness[i][1]):npp[i] += 1 # j支配 i，np+1set_sp.append(temp) # i支配 j，將 j 加入 i 的支配解集里if npp[i] == 0:fronts[0].append(i) # 個體序號rank[i] = 1 # Pareto前沿面 第一層級i = 0while len(fronts[i]) > 0:temp = []for j in range(len(fronts[i])):a = 0while a < len(set_sp[fronts[i][j]]):npp[set_sp[fronts[i][j]][a]] -= 1if npp[set_sp[fronts[i][j]][a]] == 0:rank[set_sp[fronts[i][j]][a]] = i + 2 # 第二層級temp.append(set_sp[fronts[i][j]][a])a = a + 1i = i + 1fronts.append(temp)del fronts[len(fronts) - 1]self.output_fronts(fronts)def output_fronts(self, fronts):# test codesum_coun = 0for kk in range(len(fronts)):sum_coun += len(fronts[kk])print(sum_coun)print(fronts)def test_fast_non_dominated_sort_error(self, objectives_fitness):#對華電小扎扎寫的非支配排序的修改和調整# https: // blog.csdn.net / qq_36449201 / article / details / 81046586fronts = [] # Pareto前沿面fronts.append([])set_sp = []npp = np.zeros(2*10)rank = np.zeros(2*10)for i in range(2 * 10):temp = []for j in range(2 * 10):if j != i:if (objectives_fitness[j][0] >= objectives_fitness[i][0] and objectives_fitness[j][1] >= objectives_fitness[i][1]) :temp.append(j)elif (objectives_fitness[i][0] > objectives_fitness[j][0] and objectives_fitness[i][1] > objectives_fitness[j][1]):npp[i] += 1 # j支配 i，np+1set_sp.append(temp) # i支配 j，將 j 加入 i 的支配解集里if npp[i] == 0:fronts[0].append(i) # 個體序號rank[i] = 1 # Pareto前沿面 第一層級i = 0while len(fronts[i]) > 0:temp = []for j in range(len(fronts[i])):a = 0while a < len(set_sp[fronts[i][j]]):npp[set_sp[fronts[i][j]][a]] -= 1if npp[set_sp[fronts[i][j]][a]] == 0:rank[set_sp[fronts[i][j]][a]] = i + 2 # 第二層級temp.append(set_sp[fronts[i][j]][a])a = a + 1i = i + 1fronts.append(temp)del fronts[len(fronts) - 1]self.output_fronts(fronts)def output_fronts(self, fronts):# test codesum_coun = 0for kk in range(len(fronts)):sum_coun += len(fronts[kk])print(sum_coun)print(fronts)def test_fast_non_dominated_sort_2(self, objectives_fitness):#對Github Haris Ali Khan寫的NSGA2的非支配排序函數的調整# coding:utf-8# Program Name: NSGA-II.py# Description: This is a python implementation of Prof. Kalyanmoy Deb's popular NSGA-II algorithm# Author: Haris Ali Khan# Supervisor: Prof. Manoj Kumar Tiwariset_sp=[[] for i in range(0, np.shape(objectives_fitness)[0])]fronts = [[]]npp=[0 for i in range(0, np.shape(objectives_fitness)[0])]rank = [0 for i in range(0, np.shape(objectives_fitness)[0])]for i in range(0, np.shape(objectives_fitness)[0]):set_sp[i]=[]# npp[i]=0for j in range(0, np.shape(objectives_fitness)[0]):if i != j:if (objectives_fitness[j][0] >= objectives_fitness[i][0] and objectives_fitness[j][1] > objectives_fitness[i][1]) or \(objectives_fitness[j][0] > objectives_fitness[i][0] and objectives_fitness[j][1] >= objectives_fitness[i][1]) or \(objectives_fitness[j][0] >= objectives_fitness[i][0] and objectives_fitness[j][1] >= objectives_fitness[i][1]):# 個體p的支配集合Sp計算set_sp[i].append(j)elif (objectives_fitness[i][0] >= objectives_fitness[j][0] and objectives_fitness[i][1] > objectives_fitness[j][1]) or \(objectives_fitness[i][0] > objectives_fitness[j][0] and objectives_fitness[i][1] >= objectives_fitness[j][1]) or \(objectives_fitness[j][0] > objectives_fitness[i][0] and objectives_fitness[j][1] > objectives_fitness[i][1]):# 被支配度Np計算# Np越大，則說明i個體越差npp[i] += 1 # j支配 i，np+1if npp[i]==0:rank[i] = 0if i not in fronts[0]:fronts[0].append(i)i = 0while(fronts[i] != []):Q=[]for p in fronts[i]:for q in set_sp[p]:npp[q] =npp[q] - 1if( npp[q]==0):rank[q]=i+1if q not in Q:Q.append(q)i = i+1fronts.append(Q)del fronts[len(fronts)-1]self.output_fronts(fronts)def test_fast_non_dominated_sort_3(self, objectives_fitness):# 參考MoeaPlat的非支配排序# https://blog.csdn.net/qq_40434430/article/details/82876572fronts = [] # Pareto前沿面fronts.append([])set_sp = []npp = np.zeros(2 * 10)rank = np.zeros(2 * 10)for i in range(2 * 10):temp = []for j in range(2 * 10):if j != i:less = 0 # y'的目標函數值小于個體的目標函數值數目equal = 0 # y'的目標函數值等于個體的目標函數值數目greater = 0 # y'的目標函數值大于個體的目標函數值數目for k in range(self.f_num):if (objectives_fitness[i][k] < objectives_fitness[j][k]):less = less + 1elif (objectives_fitness[i][k] == objectives_fitness[j][k]):equal = equal + 1else:greater = greater + 1if (less == 0 and equal != self.f_num):npp[i] += 1 # j支配 i，np+1elif (greater == 0 and equal != self.f_num):temp.append(j)set_sp.append(temp) # i支配 j，將 j 加入 i 的支配解集里if npp[i] == 0:fronts[0].append(i) # 個體序號rank[i] = 1 # Pareto前沿面 第一層級i = 0while len(fronts[i]) > 0:temp = []for j in range(len(fronts[i])):a = 0while a < len(set_sp[fronts[i][j]]):npp[set_sp[fronts[i][j]][a]] -= 1if npp[set_sp[fronts[i][j]][a]] == 0:rank[set_sp[fronts[i][j]][a]] = i + 2 # 第二層級temp.append(set_sp[fronts[i][j]][a])a = a + 1i = i + 1fronts.append(temp)del fronts[len(fronts) - 1]self.output_fronts(fronts)if __name__ == '__main__':test_class=Test_class(f_num=2)# test zjh 普通print("測試1：普通")test_class.test_fast_non_dominated_sort_1(test_class.objectives_fitness_zjh)test_class.test_fast_non_dominated_sort_2(test_class.objectives_fitness_zjh)test_class.test_fast_non_dominated_sort_3(test_class.objectives_fitness_zjh)test_class.test_fast_non_dominated_sort_error(test_class.objectives_fitness_zjh)# test ZDT3 正確print("測試2：正確，重點看錯誤的函數test_fast_non_dominated_sort_error")test_class.test_fast_non_dominated_sort_1(test_class.objectives_fitness_20)test_class.test_fast_non_dominated_sort_2(test_class.objectives_fitness_20)test_class.test_fast_non_dominated_sort_3(test_class.objectives_fitness_20)test_class.test_fast_non_dominated_sort_error(test_class.objectives_fitness_20)# test ZDT6 錯誤print("測試3：ZDT6 錯誤，重點看錯誤的函數test_fast_non_dominated_sort_error")test_class.test_fast_non_dominated_sort_1(test_class.objectives_fitness_8)test_class.test_fast_non_dominated_sort_2(test_class.objectives_fitness_8)test_class.test_fast_non_dominated_sort_3(test_class.objectives_fitness_8)test_class.test_fast_non_dominated_sort_error(test_class.objectives_fitness_8)

總結

這里寫的測試函數，客觀來說還是不夠好的，因為是采用窮舉的方式來驗證小于、等于、大于的關系，兩個目標函數的時候，這樣寫還是可以湊合的，但是如果是5個目標，這個顯然就不可以了。所以增加了test_fast_non_dominated_sort_3()函數，正確的做法應該是參考mooplatm中，排除的寫法，可以適用于多個目標，而不僅僅局限于兩個目標。

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