粒子群算法的寻优算法
粒子群算法（ Particle Swarm Optimization, PSO）最早是由Eberhart和Kennedy于1995年提出，它的基本概念源于对鸟群觅食行为的研究。设想这样一个场景:一群鸟在随机搜寻食物，在这个区域里只有一块食物，所有的鸟都不知道食物在哪里，但是它们知道当前的位置离食物还有多远。那么找到食物的最优策略是什么呢?最简单有效的就是搜寻目前离食物最近的鸟的周围区域。
PSO算法就从这种生物种群行为特性中得到启发并用于求解优化问题。在PSO中，每个优化问题的潜在解都可以想象成d维搜索空间上的一个点，我们称之为“粒子”（Particle），所有的粒子都有一个被目标函数决定的适应值(Fitness Value )，每个粒子还有一个速度决定他们飞翔的方向和距离，然后粒子们就追随当前的最优粒子在解空间中搜索。Reynolds对鸟群飞行的研究发现。鸟仅仅是追踪它有限数量的邻居但最终的整体结果是整个鸟群好像在一个中心的控制之下.即复杂的全局行为是由简单规则的相互作用引起的。
大致流程：
由n个粒子组成的群体对Q维（就是每个粒子的维数）空间进行搜索。每个粒子表示为：xi＝（xi1,xi2,xi3,...,xiQ），每个粒子对应的速度可以表示为vi=(vi1,vi2,vi3,....,viQ)，每个粒子在搜索时要考虑两个因素：
1。自己搜索到的历史最优值 pi ，pi=(pi1,pi2,....,piQ)，i=1,2,3,....,n。
2。全部粒子搜索到的最优值pg，pg=(pg1,pg2,....,pgQ)，注意这里的pg只有一个。
粒子群算法的位置速度更新公式：
w是保持原来速度的系数，所以叫做惯性权重。
c1是粒子跟踪自己历史最优值的权重系数，它表示粒子自身的认识，所以叫“认知”。通常设置为2。
c2是粒子跟踪群体最优值的权重系数，它表示粒子对整个群体知识的认识，所以叫做“社会知识”，经常叫做“社会”。通常设置为2。
η、ξ是[0,1]区间内均匀分布的随机数。
r是对位置更新的时候，在速度前面加的一个系数，这个系数我们叫做约束因子。通常设置为1。
代码：
%清空运行环境
clc
clear
%设置速度更新参数
c1=1.49445;
c2=1.49445;
maxgen=300; %迭代次数
sizepop=20; %种群规模
%个体和速度最大值最小值
popmax=2;
popmin=-2;
Vmax=0.5;
Vmin=-0.5;
figure;
%种群初始化
for i=1:sizepop
%随机产生一个种群
pop(i,:)=2*rands(1,2); %初始化粒子
V(i,:)=0.5*rands(1,2); %初始化速度
%计算粒子适应度值
fitness(i)=fun(pop(i,:));
end
%寻找出极值
[bestfitness bestindex]=max(fitness);
zbest=pop(bestindex,:); %群体极值位置
gbest=pop; %个体极值位置
fitnessgbest=fitness; %个体极值适应度值
fitnesszbest=bestfitness; %群体极值适应度值
%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,:));
if V(j,1)>Vmax
V(j,1)=Vmax;
end
if V(j,2)>Vmax
V(j,2)=Vmax;
end
if V(j,1)<Vmin
V(j,1)=Vmin;
end
if V(j,2)<Vmin
V(j,2)=Vmin;
end
%位置更新
pop(j,:)=pop(j,:)+0.5*V(j,:);
if pop(j,1)>popmax
pop(j,1)=popmax;
end
if pop(j,2)>popmax
pop(j,2)=popmax;
end
if pop(j,1)<popmin
pop(j,1)=popmin;
end
if pop(j,2)<popmin
pop(j,2)=popmin;
end
%新粒子适应度值
fitness(j)=fun(pop(j,:));
end
%个体极值和群体极值适应度值
for j=1:sizepop
%个体极值更新
if fitness(j)>fitnessgbest(j)
gbest(j,:)=pop(j,:);
fitnessgbest(j)=fitness(j);
end
%群体极值更新
if fitness(j)>fitnesszbest
zbest=pop(j,:);
fitnesszbest=fitness(j);
end
end
result(i)=fitnesszbest;
end
%画出每代最优化个体适应度值
plot(result)
title('最优个体适应度值','fontsize',12);
xlabel('进化代数','fontsize',12);
ylabel('适应度值','fontsize',12);
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