差分進化算法

差分進化算法(DE)是一種新興的進化計算技術，它是由 Storn 等人于1995年提出，其最初的設想是用于解決切比雪夫多項式問題，後來成為解決複雜優化問題的有效技術。

    差分進化算法的操作如下:

    (1)初始化

    (2)變異

    (3)交叉

    (4)選擇

    (5)邊界條件處理

   差分進化算法流程圖:

  matlab仿真執行個體:

  求函數f(x,y)=3cos(xy)+x+y的最小值，其中x的取值範圍為[-4,4], y的取值範圍為[-4,4](多個局部極值的函數)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%差分進化算法求函數極值%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%初始化%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
clc;
NP=20;     %種群數量
D=2;       %變量的維數
G=100;     %最大進化代數
F=0.5;     %變異算子
CR=0.1;    %交叉算子
Xs=4;      %上限
Xx=-4;     %下限

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%賦初值%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=zeros(D,NP);       %初始種群
v=zeros(D,NP);       %變異種群
u=zeros(D,NP);       %選擇種群
x=rand(D,NP)*(Xs-Xx)+Xx;         %賦初值

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%計算目标函數%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for m=1:NP
   Ob(m)=func2(x(:,m));
end
trace(1)=min(Ob);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%差分進化循環%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for gen=1:G
     %%%%%%%%%%%%%%%%%%%%%%變異操作%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     %%%%%%%%%%%%%%%%%%%%%%%%r1,r2,r3和m互不相同%%%%%%%%%%%%%%%%%%%%%%
     for m=1:NP
        r1=randint(1,1,[1,NP]);
        while(r1==m)
          r1=randint(1,1,[1,NP]);
        end
         r2=randint(1,1,[1,NP]);
         while(r2==m)|(r2==r1)
          r2=randint(1,1,[1,NP]);
         end
        r3=randint(1,1,[1,NP]);
         while((r3==m)|(r3==r1)|(r3==r2))
          r3=randint(1,1,[1,NP]);
         end
         v(:,m)=x(:,r1)+F*(x(:,r2)-x(:,r3));
     end
     
     %%%%%%%%%%%%%%%%%%%%%%交叉操作%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     r=randint(1,1,[1,NP]);
     for n=1:D
         cr=rand(1);
         if(cr<CR)|(n==r)
           u(n,:)=v(n,:);
         else
            u(n,:)=x(n,:);
         end
     end
     
     %%%%%%%%%%%%%%%%%%%%%%邊界條件的處理%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     %%%%%%%%%%%%%%%%%%%%%%邊界吸收%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     for n=1:D
       for m=1:NP
           if u(n,m)<Xx
               u(n,m)=Xx;
           end
           if u(n,m)>Xs
               u(n,m)=Xs;
           end
       end
     end
     %%%%%%%%%%%%%%%%%%%%%%選擇操作%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     for m=1:NP
         Ob1(m)=func2(u(:,m));
     end
     
    for m=1:NP
        if Ob1(m)<Ob(m)      %小于先前的目标值
            x(:,m)=u(:,m);
        end
    end
    for m=1:NP
       Ob(m)=func2(x(:,m));
    end
    trace(gen+1)=min(Ob);
end
    [SortOb,Index]=sort(Ob);
    x=x(:,Index);
    X=x(:,1);          %最優變量
    Y=min(Ob);         %最優值
   disp('最優變量');
   disp(X);
   disp('最優值');
   disp(Y);
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%畫圖%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  figure
  plot(trace);
  %plot(X,Y,'-ro');
  xlabel('疊代次數');
  ylabel('目标函數值');
  title('DE目标函數曲線');
  

  
           

%%%%%%%%%%%%%%%%%%%%%%%%%%%%适應度函數計算%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  function value=func2(x)
  value=3*cos(x(1)*x(2))+x(1)+x(2);
  end
           

運作效果如下: