模拟退火算法（Simulated Annealing，SA）MATLAB案例详细解析

一、理论

模拟退火算法（SA）、遗传算法（GA）、布谷鸟算法（CS）、人工蜂群算法（ABC）学习笔记—附MATLAB注释代码

二、模拟退火算法应用于函数优化 1.流程图

SAToFunction流程图 2.代码解析

clear all; %清除所有变量
close all; %清图
clc ;      %清屏
D = 10;
Xs = 20;%%函数上限
Xx =-20;%%函数下限
%%%%%%%%%%%%%%%%%%%%冷却表参数%%%%%%%%%%%%%%%%%%%%%%%%%%
L= 200;    %马尔可夫链长度
K = 0.998;  %衰减参数
S = 0.01;   %步长因子，就是每次解变化的大小
T = 100;    % 初始温度
YZ = 1e-8;  %容差
P = 0;      %Metropolis过程接受点

%%%%%%%%%%%%%%%%%随机选点初值设定%%%%%%%%%%%%%%%%%%%5
PreX = rand(D,1)*(Xs-Xx)+Xx;%初始化矩阵D × 1的解
PreBestX = PreX;%用于保存上一个最优的解
PreX = rand(D,1)*(Xs-Xx)+Xx;
BestX = PreX;%用于保存当前最优解
%%%%%%%%%%%%%%%%%每迭代一次退火一次（降温），直到满足迭代条件为止%%%%%%%%%
deta = abs(func1(BestX)-func1(PreBestX));%能量差值，适应度差值
while (deta>YZ) && (T>0.001)
    T = K*T;%%降温
    %%%%%%%%%%%%在当前温度T下迭代次数%%%%%%%%%%%%
    for i = 1:L
        %%%%%%%%%%%%在此点附近随机选择下一点%%%%%%%%%%%%%%%
        %%%%%%%%%%%%根据当前条件产生一个新解%%%%%%%%%%%%%%%%%
        NextX  = PreX + S*(rand(D,1) * (Xs-Xx)+Xx);
        %%%%%%%%%%%%%%%边界条件处理%%%%%%%%%%%%%
        %%%%%%%%%%%%%%因为上一行代码是随机产生的数，判断产生的数，是否超过定义域范围，超过则重新产生一个数%%%%%%
        for ii = 1:D
            if NextX(ii) > Xs | NextX(ii) < Xx
                NextX(ii) = PreX(ii) + S* (rand * (Xs-Xx)+Xx);
            end
        end
    %%%%%%%%%%%%%%是否全局最优解%%%%%%%%%%%%%%%%%%%%
    if (func1(BestX) > func1(NextX))
        %%%%%%%%%%%%%%%%%%%%保留上一个最优解%%%%%%%%%%%%%%5
        PreBestX = BestX;
        %%%%%%%%%%%%%%%%%此为新的最优解%%%%%%%%%%%%%%%%%
        BestX  = NextX;
    end
        %%%%%%%%%%%%%%%Metropolis过程%%%%%%%%%%%%%%%%%%%
    if(func1(PreX) - func1(NextX)>0)
        %%%%%%%%%%%%%%接受新解%%%%%%%%%%%%%
        PreX = NextX;
        P = P+1;
    else
        %%%%%%%%%%%%%%%%%%%求状态2接受的概率接受新解%%%%
        %%%%%%%%%%%%%%%%%%%随着越来越接近最优值的时候，接受较差的状态的概率越低%%%%%
        changer = -1 * (func1(NextX)-func1(PreX))/T;
        p1 = exp(changer);
        %%%%%%%%%%%%%%%%%%以一定概率接受较差的解%%%%%%%%%%%%%%%%%
        if p1 >rand
            PreX = NextX;
            P =P+1;
        end
    end
    trace(P+1) = func1(BestX);
    
    end
    deta = abs(func1(BestX) - func1(PreBestX));
end

%最小值的解
BestX
%解的适应度
func1(BestX)
figure
plot(trace(2:end))
xlabel('迭代次数')
ylabel('目标函数值')
title('适应度进化曲线')

%%%%%%%%%%%%%%%%%%%适应度函数%%%%%%%%%%%%%%

function result = func1(x)
summ = sum(x.^2);
result = summ;
end

3.运行计算结果

• 适应度变化曲线

• 打印解

二、模拟退火算法应用于旅行商问题（TSP） 1.流程图

2.代码解析

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 1004;
    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 2367;
    3394 2643;
    3439 3201;
    2935 3240;
    3140 3550;
    2545 2357;
    2778 2826;
    2370 2975
    ];%31个省会坐标
    
n=size(C,1); %TSP问题的规模，即城市数目
T=100*n;     %初始温度
L=100;       %马尔科夫链的长度
K=0.99;      %衰减参数

%%%城市坐标结构体%%%%%%%
city=struct([]);

for i=1:n
    city(i).x=C(i,1);
    city(i).y=C(i,2);
end

l=1;        %统计迭代次数
len(l)=func5(city,n); %每次迭代后路线的长度

figure(1);

while T>0.001
    %%%%%%%%%%%%%%%多次迭代扰动，温度降低前多次试验%%%%%%%%
    for i=1:L
        %%%%%%%%%%%%%%%计算原路线总距离%%%%%%%%%
        len1=func5(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);
        p2=floor(1+n*rand);
        while p1==p2
            p1=floor(1+n*rand);
            p2=floor(1+n*rand);
        end
        tmp_city=city;
        %%交换元素
        tmp=tmp_city(p1);
        tmp_city(p1)=tmp_city(p2);
        tmp_city(p2)=tmp;
        %%%%%%%%%%%%%%%计算新路线总距离%%%%%%%%%
        len2=func5(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，用新路线替代旧路线%%%%%%%%%
        if delta_erand()
                city=tmp_city;
            end
        end
    end
    l=l+1;          %迭代次数加1
    
    %%%%%%%%%%%%%%%计算新路线的距离%%%%%%%%%
    len(l)=func5(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))]);%%num2str将数字转为字符数组
    hold off;
    pause(0.005);
end

figure(2)
plot(len);
xlabel('迭代次数');
ylabel('目标函数值');
title('适应度进化曲线');

%计算距离的函数
function len=func5(city,n)
len=0;
for i=1:n-1
    len=len+sqrt((city(i).x-city(i+1).x)^2+(city(i).y-city(i+1).y)^2);
end
len=len+sqrt((city(n).x-city(1).x)^2+(city(n).y-city(1).y)^2);
end

3.运行计算结果

• 最短路径输出图

• 适应度收敛图