Greedy Randomized Adaptive Search 算法超详细解析,附代码实现TSP问题求解

这两天刚好看到这个算法,然后就写一写吧。贪心随机自适应搜索虽然算是一个比较简单的启发式,但是效果也非常不错的。

01 概述

Greedy Randomized Adaptive Search,贪婪随机自适应搜索(GRAS),是组合优化问题中的多起点元启发式算法。

在算法的每次迭代中,主要由两个阶段组成:构造(construction)和局部搜索( local search)。

构造(construction)阶段主要用于生成一个可行解,而后该初始可行解会被放进局部搜索进行邻域搜索,直到找到一个局部最优解为止。

02 整体框架

如上面所说,其实整一个算法的框架相对于其他算法来说还算比较简单明了,大家可以先看以下整体的伪代码:

(GRAS伪代码)

GRAS主要由两部分组成:

1) Greedy_Randomized_Construction:在贪心的基础上,加入一定的随机因素,构造初始可行解。

2) Local Search:对上面构造的初始可行解进行邻域搜索,直到找到一个局部最优解。

然后再多说两句:

Repair是什么鬼?有时候由于随机因素的加入,Greedy_Randomized_Construction阶段生成的解不一定都是可行解,所以为了保证下一步的Local Search能继续进行,加入repair算子,对解进行修复,保证其可行。

不是说自适应(Adaptive)吗?我怎么没看到Adaptive 的过程? 别急,这个后面具体举例的时候会详细讲到。

03 举个例子说明

为了大家能更加深入理解该算法,我们举一个简单的例子来为大家详细讲解算法的流程。

假如有一个有限的集合E = {1, 2, 3, ……, n}。可行解的集合

,(我知道你们想问

是什么,由E的所有子集作为元素构成的集合。)目标函数

求目标函数的最小值以及对应的解。

好了,相信大家都看懂上面的问题了(看不懂也别问我,摊手)。对于上述问题,我们来一步一个脚印用GRAS来求解之,来,跟紧小编的脚步……

强调了很多次,GRAS由两部分组成:Greedy_Randomized_Construction和Local Search,所以,在求解具体问题的时候,完成这两部分的设计,然后按照第二节所示的框架搭起来就可以。

3.1 Greedy_Randomized_Construction

这里还是老规矩,先上伪代码给大家看看,然后我们再进行讲解,毕竟对于算法来说,伪代码的作用不言而喻。

Greedy_Randomized_Construction

第1行,一开始解是一个空集。

第2行,初始化候选元素的集合,这里候选元素是指能放进Solution的元素(也就是目前Solution里面没有的),比如1,2,3……。

第3行,对候选集合的每个元素进行评估,计算将元素x放入Solution会导致目标函数f改变量delta_x。

第5行,根据delta_x对各个元素排序,选取部分较好的候选元素组成RCL表(贪心性体现在这里)。

第6行,随机在RCL中选取一个元素放进Solution。(算法的随机性)

第8、9行,更新候选元素集合,然后对每个元素进行重新评估计算delta值。(算法的自适应性体现在这里)

相信经过上面如此详细的介绍,大家都懂了吧!

3.2 Local Search

关于Local Search方面的内容,相信大家学习heuristic这么久了,就不用我多说什么了吧:

(Local Search)

简单看一下伪代码即可,主要是邻域算子的设计,然后就是在邻域里面进行搜索,找到一个局部最优解为止。

然后关于邻域搜索,有best-improving or first-improving strategy 两种策略,这个下次有时间出个专题给大家讲明白一些相关概念吧。

04 再论Greedy_Randomized_Construction

前面我们说了,Greedy_Randomized_Construction用于生成初始解,既然是Greedy_Randomized两个结合体,那么肯定就有一个权重分配的问题。

即,是Greedy成分多一点呢?还是Randomized成分多一点好呢?因此,为了控制这两个小老弟的权重,防止某个家伙在该过程中用力过猛导致解不那么好的情况,我们引入一个参数α:

其他部分就不再多说,可以看到,上面的α参数主要是控制RCL(什么是RCL回头去看!)的长度:

当α=0时,纯贪心,只能选取最优的候选元素。

当α=1时,纯随机,所有候选元素都可随机选。

05 代码实现

由于小编精力有限,就不从头写一遍了,从GitHub上找了一个感觉还不错的算法给大家,也是求解TSP问题的。

不过说实在的,python写算法的速度是很慢的,无论是速度还是算法架构等方面都不推荐大家用matlab或者python这种脚本性的语言写大型优化算法。

运行结果如下:

(Berlin52)

代码算例以及相关运行结果请移步留言区。

############################################################################

# Created by: Prof. Valdecy Pereira, D.Sc.
# UFF - Universidade Federal Fluminense (Brazil)
# email:  valdecy.pereira@gmail.com
# Course: Metaheuristics
# Lesson: Local Search-GRASP

# Citation: 
# PEREIRA, V. (2018). Project: Metaheuristic-Local_Search-GRASP, File: Python-MH-Local Search-GRASP.py, GitHub repository: <https://github.com/Valdecy/Metaheuristic-Local_Search-GRASP>

############################################################################

# Required Libraries
import pandas as pd
import random
import numpy  as np
import copy
import os
from matplotlib import pyplot as plt 

# Function: Tour Distance
def distance_calc(Xdata, city_tour):
    distance = 0
    for k in range(0, len(city_tour[0])-1):
        m = k + 1
        distance = distance + Xdata.iloc[city_tour[0][k]-1, city_tour[0][m]-1]            
    return distance

# Function: Euclidean Distance 
def euclidean_distance(x, y):       
    distance = 0
    for j in range(0, len(x)):
        distance = (x.iloc[j] - y.iloc[j])**2 + distance   
    return distance**(1/2) 

# Function: Initial Seed
def seed_function(Xdata):
    seed = [[],float("inf")]
    sequence = random.sample(list(range(1,Xdata.shape[0]+1)), Xdata.shape[0])
    sequence.append(sequence[0])
    seed[0] = sequence
    seed[1] = distance_calc(Xdata, seed)
    return seed

# Function: Build Distance Matrix
def buid_distance_matrix(coordinates):
    Xdata = pd.DataFrame(np.zeros((coordinates.shape[0], coordinates.shape[0])))
    for i in range(0, Xdata.shape[0]):
        for j in range(0, Xdata.shape[1]):
            if (i != j):
                x = coordinates.iloc[i,:]
                y = coordinates.iloc[j,:]
                Xdata.iloc[i,j] = euclidean_distance(x, y)        
    return Xdata

# Function: Tour Plot
def plot_tour_distance_matrix (Xdata, city_tour):
    m = Xdata.copy(deep = True)
    for i in range(0, Xdata.shape[0]):
        for j in range(0, Xdata.shape[1]):
            m.iloc[i,j] = (1/2)*(Xdata.iloc[0,j]**2 + Xdata.iloc[i,0]**2 - Xdata.iloc[i,j]**2)    
    m = m.values
    w, u = np.linalg.eig(np.matmul(m.T, m))
    s = (np.diag(np.sort(w)[::-1]))**(1/2) 
    coordinates = np.matmul(u, s**(1/2))
    coordinates = coordinates.real[:,0:2]
    xy = pd.DataFrame(np.zeros((len(city_tour[0]), 2)))
    for i in range(0, len(city_tour[0])):
        if (i < len(city_tour[0])):
            xy.iloc[i, 0] = coordinates[city_tour[0][i]-1, 0]
            xy.iloc[i, 1] = coordinates[city_tour[0][i]-1, 1]
        else:
            xy.iloc[i, 0] = coordinates[city_tour[0][0]-1, 0]
            xy.iloc[i, 1] = coordinates[city_tour[0][0]-1, 1]
    plt.plot(xy.iloc[:,0], xy.iloc[:,1], marker = 's', alpha = 1, markersize = 7, color = 'black')
    plt.plot(xy.iloc[0,0], xy.iloc[0,1], marker = 's', alpha = 1, markersize = 7, color = 'red')
    plt.plot(xy.iloc[1,0], xy.iloc[1,1], marker = 's', alpha = 1, markersize = 7, color = 'orange')
    return

# Function: Tour Plot
def plot_tour_coordinates (coordinates, city_tour):
    coordinates = coordinates.values
    xy = pd.DataFrame(np.zeros((len(city_tour[0]), 2)))
    for i in range(0, len(city_tour[0])):
        if (i < len(city_tour[0])):
            xy.iloc[i, 0] = coordinates[city_tour[0][i]-1, 0]
            xy.iloc[i, 1] = coordinates[city_tour[0][i]-1, 1]
        else:
            xy.iloc[i, 0] = coordinates[city_tour[0][0]-1, 0]
            xy.iloc[i, 1] = coordinates[city_tour[0][0]-1, 1]
    plt.plot(xy.iloc[:,0], xy.iloc[:,1], marker = 's', alpha = 1, markersize = 7, color = 'black')
    plt.plot(xy.iloc[0,0], xy.iloc[0,1], marker = 's', alpha = 1, markersize = 7, color = 'red')
    plt.plot(xy.iloc[1,0], xy.iloc[1,1], marker = 's', alpha = 1, markersize = 7, color = 'orange')
    return

# Function: Rank Cities by Distance
def ranking(Xdata, city = 0):
    rank = pd.DataFrame(np.zeros((Xdata.shape[0], 2)), columns = ['Distance', 'City'])
    for i in range(0, rank.shape[0]):
        rank.iloc[i,0] = Xdata.iloc[i,city]
        rank.iloc[i,1] = i + 1
    rank = rank.sort_values(by = 'Distance')
    return rank

# Function: RCL
def restricted_candidate_list(Xdata, greediness_value = 0.5):
    seed = [[],float("inf")]
    sequence = []
    sequence.append(random.sample(list(range(1,Xdata.shape[0]+1)), 1)[0])
    for i in range(0, Xdata.shape[0]):
        count = 1
        rand = int.from_bytes(os.urandom(8), byteorder = "big") / ((1 << 64) - 1)
        if (rand > greediness_value and len(sequence) < Xdata.shape[0]):
            next_city = int(ranking(Xdata, city = sequence[-1] - 1).iloc[count,1])
            while next_city in sequence:
                count = np.clip(count+1,1,Xdata.shape[0]-1)
                next_city = int(ranking(Xdata, city = sequence[-1] - 1).iloc[count,1])
            sequence.append(next_city)
        elif (rand <= greediness_value and len(sequence) < Xdata.shape[0]):
            next_city = random.sample(list(range(1,Xdata.shape[0]+1)), 1)[0]
            while next_city in sequence:
                next_city = int(random.sample(list(range(1,Xdata.shape[0]+1)), 1)[0])
            sequence.append(next_city)
    sequence.append(sequence[0])
    seed[0] = sequence
    seed[1] = distance_calc(Xdata, seed)
    return seed

# Function: 2_opt
def local_search_2_opt(Xdata, city_tour):
    tour = copy.deepcopy(city_tour)
    best_route = copy.deepcopy(tour)
    seed = copy.deepcopy(tour)  
    for i in range(0, len(tour[0]) - 2):
        for j in range(i+1, len(tour[0]) - 1):
            best_route[0][i:j+1] = list(reversed(best_route[0][i:j+1]))           
            best_route[0][-1]  = best_route[0][0]                          
            best_route[1] = distance_calc(Xdata, best_route)           
            if (best_route[1] < tour[1]):
                tour[1] = copy.deepcopy(best_route[1])
                for n in range(0, len(tour[0])): 
                    tour[0][n] = best_route[0][n]          
            best_route = copy.deepcopy(seed) 
    return tour

# Function: GRASP
def greedy_randomized_adaptive_search_procedure(Xdata, city_tour, iterations = 50, rcl = 25, greediness_value = 0.5):
    count = 0
    best_solution = copy.deepcopy(city_tour)
    while (count < iterations):
        rcl_list = []
        for i in range(0, rcl):
            rcl_list.append(restricted_candidate_list(Xdata, greediness_value = greediness_value))
        candidate = int(random.sample(list(range(0,rcl)), 1)[0])
        city_tour = local_search_2_opt(Xdata, city_tour = rcl_list[candidate])
        while (city_tour[0] != rcl_list[candidate][0]):
            rcl_list[candidate] = copy.deepcopy(city_tour)
            city_tour = local_search_2_opt(Xdata, city_tour = rcl_list[candidate])
        if (city_tour[1] < best_solution[1]):
            best_solution = copy.deepcopy(city_tour) 
        count = count + 1
        print("Iteration =", count, "-> Distance =", best_solution[1])
    print("Best Solution =", best_solution)
    return best_solution

######################## Part 1 - Usage ####################################

X = pd.read_csv('Python-MH-Local Search-GRASP-Dataset-01.txt', sep = '\t') #17 cities = 1922.33
seed = seed_function(X)
lsgrasp = greedy_randomized_adaptive_search_procedure(X, city_tour = seed, iterations = 5, rcl = 5, greediness_value = 0.5)
plot_tour_distance_matrix(X, lsgrasp) # Red Point = Initial city; Orange Point = Second City # The generated coordinates (2D projection) are aproximated, depending on the data, the optimum tour may present crosses.

Y = pd.read_csv('Python-MH-Local Search-GRASP-Dataset-02.txt', sep = '\t') # Berlin 52 = 7544.37
X = buid_distance_matrix(Y)
seed = seed_function(X)
lsgrasp = greedy_randomized_adaptive_search_procedure(X, city_tour = seed, iterations = 10, rcl = 15, greediness_value = 0.5)
plot_tour_coordinates (Y, lsgrasp) # Red Point = Initial city; Orange Point = Second City

原文发布于微信公众号 - 程序猿声(ProgramDream)

原文发表时间:2019-06-05

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