模拟退火算法简介及求TSP问题(MATLAB)

1. 模拟退火算法简介

网上和书本上有太多关于模拟退火算法的介绍，写的都不错，大家可以参考。
例如：https://blog.csdn.net/huahua19891221/article/details/81737053

2. 模拟退火算法解决TSP问题

2.1 问题描述如下：

2.2 仿真过程描述如下：

2.3 仿真代码如下：

%%%%%%%%%%%%%%%%%%%%%%模拟退火算法解决TSP问题%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%初始化%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;                      %清除所有变量
close all;                      %清图
clc;                            %清屏
C=[1304 2312;3639 1315;4177 2244;3712 1399;3488 1535;3326 1556;...
    3238 1229;4196 1044;4312  790;4386  570;3007 1970;2562 1756;...
    2788 1491;2381 1676;1332  695;3715 1678;3918 2179;4061 2370;...
    3780 2212;3676 2578;4029 2838;4263 2931;3429 1908;3507 2376;...
    3394 2643;3439 3201;2935 3240;3140 3550;2545 2357;2778 2826;...
    2370 2975];                  %31个省会城市坐标
n=size(C,1);                     %TSP问题的规模,即城市数目
T=100*n;                         %初始温度
L=100;                           %马可夫链长度
K=0.99;                          %衰减参数
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%城市坐标结构体%%%%%%%%%%%%%%%%%%%%%%%%%%
city=struct([]);                %结构体变量，类似python中的字典
for i=1:n                       %city（i）的值为第i座城市的坐标
    city(i).x=C(i,1);
    city(i).y=C(i,2);
end
l=1;                             %统计迭代次数
len(l)=func3(city,n);            %每次迭代后的路线长度
% figure(1); 
while T>0.001                    %停止迭代温度
    %%%%%%%%%%%%%%%%多次迭代扰动，温度降低之前多次实验%%%%%%%%%%%%%%%
    for i=1:L            
        %%%%%%%%%%%%%%%%%%%计算原路线总距离%%%%%%%%%%%%%%%%%%%%%%%%%
        len1=func3(city,n);         
        %%%%%%%%%%%%%%%%%%%%%%%%%产生随机扰动%%%%%%%%%%%%%%%%%%%%%%%
        %%%%%%%%%%%%%%%%随机置换两个不同的城市的坐标%%%%%%%%%%%%%%%%%
        p1=floor(1+n*rand());           %朝负无穷方向取整，如floor(-1.3)=-2，ceil相反
        p2=floor(1+n*rand());            %这是书上的方法
        while p1==p2
            p1=floor(1+n*rand());
            p2=floor(1+n*rand());
        end
        tmp_city=city;
        tmp=tmp_city(p1);
        tmp_city(p1)=tmp_city(p2);
        tmp_city(p2)=tmp;

        %%%%%%%%%%%%%%%%%%%%%%%%计算新路线总距离%%%%%%%%%%%%%%%%%%%%
        len2=func3(tmp_city,n);     
        %%%%%%%%%%%%%%%%%%新老距离的差值，相当于能量%%%%%%%%%%%%%%%%%
        delta_e=len2-len1;
        %%%%%%%%%%%%新路线好于旧路线，用新路线代替旧路线%%%%%%%%%%%%%%  
        if delta_e<0        
            city=tmp_city;
        else
            %%%%%%%%%%%%%%%%%%以概率选择是否接受新解%%%%%%%%%%%%%%%%%
            if exp(-delta_e/T)>rand()
                city=tmp_city;      
            end
        end
    end
    l=l+1;
    %%%%%%%%%%%%%%%%%%%%%%%%%计算新路线距离%%%%%%%%%%%%%%%%%%%%%%%%%%
    len(l)=func3(city,n); 
    %%%%%%%%%%%%%%%%%%%%%%%%%%%温度不断下降%%%%%%%%%%%%%%%%%%%%%%%%%%
     T=T*K;   
%     for i=1:n-1
%         plot([city(i).x,city(i+1).x],[city(i).y,city(i+1).y],'bo-');
%         hold on;
%     end
%     plot([city(n).x,city(1).x],[city(n).y,city(1).y],'ro-');
%     title(['优化最短距离:',num2str(len(l))]);
%     hold off;
%     pause(0.005);
end
figure(1);
for i=1:n-1
    plot([city(i).x,city(i+1).x],[city(i).y,city(i+1).y],'bo-');
    hold on;
end
plot([city(n).x,city(1).x],[city(n).y,city(1).y],'ro-');
title(['优化最短距离:',num2str(len(l))])
hold off;
figure(2);
plot(len)
% plot(C(:,1),C(:,2),'bo-')以C的第1列为横坐标，第二列为纵坐标
xlabel('迭代次数')
ylabel('目标函数值')
title('适应度进化曲线')

2.4 仿真结果

由于TSP问题是NP问题，所以每次求解的结果可能不一样，本次仿真结果所求最短路径如下所示：

适应度进化曲线如下所示：