First, the basic flow algorithm
1.1 Algorithm Overview
Particle swarm optimization (PSO, particle swarm optimization) algorithm is the field of computational intelligence, swarm intelligence optimization algorithm of ant colony algorithm in addition, fish algorithm, the algorithm was first proposed in 1995 by Kennedy and Eberhart, the algorithm Research on birds of prey from the issue.
Algorithm steps:
(1) initializes all particles, including their location and speed, the optimum individual pBest history to the current position, and most of the current population of individuals as gBest;
(2) the evolution of each generation, each particle is calculated the fitness function value;
(3), be compared to its fitness value of each particle and its best position pbest experienced, if preferred, it is the best as the current position pBest;
(. 4) for each particle, and its fitness value are compared and the best global position gbest experience, if preferred, the reset gBest index number;
speed (5) and a position change of microparticles;
(6) fails to reach end condition (usually a good fitness is enough or reach a pre-set maximum generation Gmax), back (2), otherwise the output gBest and ends.
In each iteration, the particles extremum update their velocity and position by individual and group extremum, the following update equations:
1.2 algorithm flow chart
Second, code implementation
PSO.m
%% 清空环境
clc
clear
%% 参数初始化
%粒子群算法中的三个参数
c1 = 1.49445;%加速因子
c2 = 1.49445;
w=0.8 %惯性权重
maxgen=1000; % 进化次s数
sizepop=200; %种群规模
Vmax=1; %限制速度围
Vmin=-1;
popmax=5; %变量取值范围
popmin=-5;
dim=10; %适应度函数维数
func=1; %选择待优化的函数，1为Rastrigin,2为Schaffer,3为Griewank
Drawfunc(func);%画出待优化的函数，只画出二维情况作为可视化输出
%% 产生初始粒子和速度
for i=1:sizepop
%随机产生一个种群
pop(i,:)=popmax*rands(1,dim); %初始种群
V(i,:)=Vmax*rands(1,dim); %初始化速度
%计算适应度
fitness(i)=fun(pop(i,:),func); %粒子的适应度
end
%% 个体极值和群体极值
[bestfitness bestindex]=min(fitness);
gbest=pop(bestindex,:); %全局最佳
pbest=pop; %个体最佳
fitnesspbest=fitness; %个体最佳适应度值
fitnessgbest=bestfitness; %全局最佳适应度值
%% 迭代寻优
for i=1:maxgen
fprintf('第%d代，',i);
fprintf('最优适应度%f\n',fitnessgbest);
for j=1:sizepop
%速度更新
V(j,:) = w*V(j,:) + c1*rand*(pbest(j,:) - pop(j,:)) + c2*rand*(gbest - pop(j,:)); %根据个体最优pbest和群体最优gbest计算下一时刻速度
V(j,find(V(j,:)>Vmax))=Vmax; %限制速度不能太大
V(j,find(V(j,:)<Vmin))=Vmin;
%种群更新
pop(j,:)=pop(j,:)+0.5*V(j,:); %位置更新
pop(j,find(pop(j,:)>popmax))=popmax;%坐标不能超出范围
pop(j,find(pop(j,:)<popmin))=popmin;
if rand>0.98 %加入变异种子，用于跳出局部最优值
pop(j,:)=rands(1,dim);
end
%更新第j个粒子的适应度值
fitness(j)=fun(pop(j,:),func);
end
for j=1:sizepop
%个体最优更新
if fitness(j) < fitnesspbest(j)
pbest(j,:) = pop(j,:);
fitnesspbest(j) = fitness(j);
end
%群体最优更新
if fitness(j) < fitnessgbest
gbest = pop(j,:);
fitnessgbest = fitness(j);
end
end
yy(i)=fitnessgbest;
end
%% 结果分析
figure;
plot(yy)
title('最优个体适应度','fontsize',12);
xlabel('进化代数','fontsize',12);ylabel('适应度','fontsize',12);
Rastrigin.m
Images:
function y = Rastrigin(x)
% Rastrigin函数
% 输入x,给出相应的y值,在x = ( 0 , 0 ,…, 0 )处有全局极小点0.
% 编制人：
% 编制日期：
[row,col] = size(x);
if row > 1
error( ' 输入的参数错误 ' );
end
y =sum(x.^2-10*cos(2*pi*x)+10);
%y =-y;Schaffer.m
Function writing:
This function has a maximum at 1 at (0, ..., 0), and therefore does not need to take the opposite number.
Images:
function y=Schaffer(x)
[row,col]=size(x);
if row>1
error('输入的参数错误');
end
y1=x(1,1);
y2=x(1,2);
temp=y1^2+y2^2;
y=0.5-(sin(sqrt(temp))^2-0.5)/(1+0.001*temp)^2;
%y=-y;Griewank.m
Images:
function y=Griewank(x)
%Griewan函数
%输入x,给出相应的y值,在x=(0,0,…,0)处有全局极小点0.
%编制人：
%编制日期：
[row,col]=size(x);
if row>1
error('输入的参数错误');
end
y1=1/4000*sum(x.^2);
y2=1;
for h=1:col
y2=y2*cos(x(h)/sqrt(h));
end
y=y1-y2+1;
%y=-y;
fun.m
For calculating a fitness function value of the particle, the choice function to be optimized, 1 Rastrigin, 2 to Schaffer, 3 to Griewank
function y = fun(x,label)
%x input 输入粒子
%y output 粒子适应度值
if label==1
y=Rastrigin(x);
elseif label==2
y=Schaffer(x);
else
y= Griewank(x);
end
fun1.m
function y = fun1(x)
%函数用于计算粒子适应度值
%x input 输入粒子
%y output 粒子适应度值
y=-20*exp(-0.2*sqrt((x(1)^2+x(2)^2)/2))-exp((cos(2*pi*x(1))+cos(2*pi*x(2)))/2)+20+exp(1);
%y=x(1)^2-10*cos(2*pi*x(1))+10+x(2)^2-10*cos(2*pi*x(2))+10;
Drawfunc.m
Draw function to be optimized, shown only as a two-dimensional case the visual output
function Drawfunc(label)
x=-5:0.05:5;%41列的向量
if label==1
y = x;
[X,Y] = meshgrid(x,y);
[row,col] = size(X);
for l = 1 :col
for h = 1 :row
z(h,l) = Rastrigin([X(h,l),Y(h,l)]);
end
end
surf(X,Y,z);
shading interp
xlabel('x1-axis'),ylabel('x2-axis'),zlabel('f-axis');
title('mesh');
end
if label==2
y = x;
[X,Y] = meshgrid(x,y);
[row,col] = size(X);
for l = 1 :col
for h = 1 :row
z(h,l) = Schaffer([X(h,l),Y(h,l)]);
end
end
surf(X,Y,z);
shading interp
xlabel('x1-axis'),ylabel('x2-axis'),zlabel('f-axis');
title('mesh');
end
if label==3
y = x;
[X,Y] = meshgrid(x,y);
[row,col] = size(X);
for l = 1 :col
for h = 1 :row
z(h,l) = Griewank([X(h,l),Y(h,l)]);
end
end
surf(X,Y,z);
shading interp
xlabel('x1-axis'),ylabel('x2-axis'),zlabel('f-axis');
title('mesh');
end
PSOMutation.m
%% 清空环境
clc
clear
%% 参数初始化
%粒子群算法中的两个参数
c1 = 1.49445;
c2 = 1.49445;
maxgen=500; % 进化次数
sizepop=100; %种群规模
Vmax=1;
Vmin=-1;
popmax=5;
popmin=-5;
%% 产生初始粒子和速度
for i=1:sizepop
%随机产生一个种群
pop(i,:)=5*rands(1,2); %初始种群
V(i,:)=rands(1,2); %初始化速度
%计算适应度
fitness(i)=fun(pop(i,:)); %染色体的适应度
end
%% 个体极值和群体极值
[bestfitness bestindex]=min(fitness);
zbest=pop(bestindex,:); %全局最佳
gbest=pop; %个体最佳
fitnessgbest=fitness; %个体最佳适应度值
fitnesszbest=bestfitness; %全局最佳适应度值
%% 迭代寻优
for i=1:maxgen
for j=1:sizepop
%速度更新
V(j,:) = V(j,:) + c1*rand*(gbest(j,:) - pop(j,:)) + c2*rand*(zbest - pop(j,:));
V(j,find(V(j,:)>Vmax))=Vmax;
V(j,find(V(j,:)<Vmin))=Vmin;
%种群更新
pop(j,:)=pop(j,:)+0.5*V(j,:);
pop(j,find(pop(j,:)>popmax))=popmax;
pop(j,find(pop(j,:)<popmin))=popmin;
if rand>0.98
pop(j,:)=rands(1,2);
end
%适应度值
fitness(j)=fun(pop(j,:));
end
for j=1:sizepop
%个体最优更新
if fitness(j) < fitnessgbest(j)
gbest(j,:) = pop(j,:);
fitnessgbest(j) = fitness(j);
end
%群体最优更新
if fitness(j) < fitnesszbest
zbest = pop(j,:);
fitnesszbest = fitness(j);
end
end
yy(i)=fitnesszbest;
end
%% 结果分析
plot(yy)
title('最优个体适应度','fontsize',12);
xlabel('进化代数','fontsize',12);ylabel('适应度','fontsize',12);
Third, the results analysis
POS.m operating results
The best individual fitness closest to zero algorithm to achieve good
Algorithm parameters:
%粒子群算法中的三个参数
加速因子
c1 向自身推进的加速权值
c2 向全局推进的加速权值
w 惯性权重
sizepop 种群规模
dim 适应度函数维数
Initial Setting: c1 = 1.49445, c2 = 1.49445, w = 0.8, sizepop = 200, dim = 10
（1）c1 = 1.49445，c2 = 1.49445，sizepop=200，dim=10，改变惯性权重的值 w
| w | 适应度平均值 |
| 0.8 | 3.581852 |
| 0.7 | 2.586893 |
| 0.6 | 2.967745 |
| 0.4 | 1.893848 |
w关系到前一速度对当前速度的影响，一般取值在0.9~0.4。因初始种群是随机产生的，当权重w固定为某一个值的时候，我们可以用多次结果取平均值的方法来处理数据。
（2）利用线性递减的函数设置w从0.9到0.4变化，观察记录这时多个结果不同
典型线性递减策略的w计算公式如下：
其中 wmax是惯性权重最大值，wmin是惯性权重最小值，t 是当前迭代次数，是个变量，
maxgen是总共迭代次数，是常量，实验中取 1000。
当初始迭代时，惯性权重w比较大，反应在速度v的计算公式上（V(j,:) = w*V(j,:) + c1*rand*(pbest(j,:) - pop(j,:)) + c2*rand*(gbest - pop(j,:))）可以看出初始迭代的时候粒子的速度比较大，具有很好的全局搜索能力，而局部搜索能力较弱。随着迭代次数的累加，w的值越来越小，粒子的速度也越来越小，此时粒子具有很好的局部搜索能力，而全局搜索能力较弱。但是由于斜率恒定，所以速度的改变总是保持同一水平。
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 平均适应度 | |
| 1 | 3.979836 | 3.979836 | 0 | 1.989918 | 2.538381 | 4.974795 | 6.964713 | 0 | 5.969754 | 4.974795 | 3.537203 |
| 2 | 0 | 0.000009 | 0.994962 | 5.969754 | 3.979836 | 0 | 0 | 4.974795 | 4.977137 | 0.995012 | 2.189151 |
| 3 | 0.994959 | 6.964713 | 4.974795 | 1.048167 | 4.974795 | 5.969754 | 0.994959 | 3.991431 | 2.984877 | 2.984877 | 3.588333 |
在迭代寻找最优的代码循环里利用公式来取惯性权重w：
%% 迭代寻优
for i=1:maxgen
fprintf('第%d代，',i);
fprintf('最优适应度%f\n',fitnessgbest);
for j=1:sizepop
%速度更新
V(j,:) = w*V(j,:) + c1*rand*(pbest(j,:) - pop(j,:)) + c2*rand*(gbest - pop(j,:)); %根据个体最优pbest和群体最优gbest计算下一时刻速度
V(j,find(V(j,:)>Vmax))=Vmax; %限制速度不能太大
V(j,find(V(j,:)<Vmin))=Vmin;
%种群更新
pop(j,:)=pop(j,:)+0.5*V(j,:); %位置更新
pop(j,find(pop(j,:)>popmax))=popmax;%坐标不能超出范围
pop(j,find(pop(j,:)<popmin))=popmin;
if rand>0.98 %加入变异种子，用于跳出局部最优值
pop(j,:)=rands(1,dim);
end
%更新第j个粒子的适应度值
fitness(j)=fun(pop(j,:),func);
end
for j=1:sizepop
%个体最优更新
if fitness(j) < fitnesspbest(j)
pbest(j,:) = pop(j,:);
fitnesspbest(j) = fitness(j);
end
%群体最优更新
if fitness(j) < fitnessgbest
gbest = pop(j,:);
fitnessgbest = fitness(j);
end
end
yy(i)=fitnessgbest;
w=wmax-(wmax-wmin)*i/1000;
end
线性微分递减策略的w的计算公式如下：
初始迭代的时候，w变化缓慢，有利于在初始迭代时寻找满足条件的局部最优值，在接近最大迭代次数时，w变化较快，在寻找到局部最优值之后能够快速地收敛逼近于全局最优值，提高运算效率。
（3）其他参数不变，只改变适应度函数维度dim：
| dim | 最优适应度 | 收敛（次） |
| 5 | 0 | 469 |
| 6 | 0 | 697 |
| 8 | 0 | 752 |
| 10 | 0 | 483 |
pop(i,:)=popmax*rands(1,dim); %初始种群
V(i,:)=Vmax*rands(1,dim); %初始化速度适应度函数维度用于初始化种群和速度，以及后面若是算法陷入局部最优时加入的变异因子。当dim越大，初始种群和速度比较大的几率高。实验取dim=5或者dim=10时迭代收敛得比较快。
（4）其他参数不变，只c1、c2改变
| 适应度 | c2=1.08324 | c2=1.49445 | c2=1.99857 |
| c1=1.08324 | 3.979836 | 0.994959 | 3.979836 |
| c1=1.49445 | 3.979836 | 1.989918 | 4.974795 |
| c1=1.99857 | 5.969763 | 2.984877 | 1.989918 |
可以看出在此目标函数中，相对三个取值中，c1取1.08324，c2取1.49445，适应度的值更接近于0。
（5）其他参数不变，改变种群规模sizepop
| sizepop | 最优适应度 | 达到最优适应度时收敛次数 | 总用时 |
| 100 | 0 | 375 | 6.041 s |
| 200 | 0 | 530 | 8.334 s |
| 300 | 0 | 115 | 12.173 s |
| 400 | 0 | 87 | 14.392 s |
| 500 | 0 | 523 | 18.376 s |
| 600 | 0 | 239 | 20.882 s |
种群规模sizepop影响算法的搜索能力和计算量。当种群规模越大时，达到最优适应度所用的时间越长，此目标函数中种群规模设置在300或400时达到最优适应度时收敛次数比较小，算法效果较好。
