Optimization methods 最优化方法

Gradient descent 梯度下降法

%梯度下降 tiduxiajing.m

function min=tiduxiajiang()

syms x1 x2 x3 x4;

f=(x1+10*x2)^2+5*(x3-x4)^2+(x2-2*x3)^4+10*(x1-x4)^4;

e=0.05;

k=0;

x0=[1,2,3,4];

x=x0;

tol=1;

df=jacobian(f)

while((tol>e)&(k<100))  

      df=-jacobian(f);

      dk=subs(df,[x1,x2,x3,x4],x);

      tol=norm(dk);

      dk=subs(dk);

      x=hjfg(x,dk) ;

      k=k+1;

end

min=subs(f,[x1,x2,x3,x4],x);

disp('采用最速梯度法求解多元函数:Min {f=(x1+10*x2)^2+5*(x3-x4)^2+(x2-2*x3)^4+10*(x1-x4)^4}的最优解：')

disp('选取初始点为：'),disp(x0)

disp('迭代次数：'),disp(k)

disp('最小点是：'),disp(x)

disp('此时的最小值为：'),disp(min) 

 

%黄金分割 huangjinfenge.m 

function m=hjfg(x,dk)
syms x1 x2 x3 x4;

f=(x1+10*x2)^2+5*(x3-x4)^2+(x2-2*x3)^4+10*(x1-x4)^4;

e=0.01;

k=0;

a=x;

b=x+dk;

tol=1;

while((tol>e)&(k<100))

    q1=a+0.328*(b-a);

    q2=a+0.618*(b-a);

    f1=subs(f,[x1,x2,x3,x4],q1);

    f2=subs(f,[x1,x2,x3,x4],q2);

    if f1>f2

         a=q1;

    else

         b=q2;

    end

   tol=norm(b-a);

end

m=(a+b)/2; 

Newton's method 牛顿法

法 

function min=newton()
syms  x1 x2 x3 x4;

f=(x1+10*x2)^2+5*(x3-x4)^2+(x2-2*x3)^4+10*(x1-x4)^4;

e=10e-4;

k=0;

x0=[1,2,3,4];

x=x0;

df=jacobian(f);

dd=subs(df,[x1,x2,x3,x4],x);

tol=norm(dd);

while((tol>e)&(k<100))

     dm1=[diff(df(1),x1),diff(df(1),x2),diff(df(1),x3),diff(df(1),x4)]';

     dm2=[diff(df(2),x1),diff(df(2),x2),diff(df(2),x3),diff(df(2),x4)]';

     dm3=[diff(df(3),x1),diff(df(3),x2),diff(df(3),x3),diff(df(3),x4)]';

     dm4=[diff(df(4),x1),diff(df(4),x2),diff(df(4),x3),diff(df(4),x4)]';

     dm=[dm1,dm2,dm3,dm4]';

     dm=subs(dm,[x1,x2,x3,x4],x);

     dk=-dm\dd';

     x=huangjinfenge(x,dk');

     dd=subs(df,[x1,x2,x3,x4],x);

     tol=norm(dd);

     k=k+1;

end

min=subs(f,[x1,x2,x3,x4],x);

disp('采用牛顿法求解多元函数:Min {f=(x1+10*x2)^2+5*(x3-x4)^2+(x2-2*x3)^4+10*(x1-x4)^4}的最优解：')

disp('选取初始点为：'),disp(x0)

disp('迭代次数：'),disp(k)

disp('最小点为：'),disp(x)

disp('此时的最小值为：'),disp(min)

 

%黄金分割

f=(x1+10*x2)^2+5*(x3-x4)^2+(x2-2*x3)^4+10*(x1-x4)^4;

内点法和内点法一维搜索

%内点法

function min=neidianfa()

e=0.005;

k=0;

t=0;

m=1;

u=0.5;

syms  x1 x2 ;

f=x1^3+x2^2+m*(1/(x1+x2-1));

df=jacobian(f);

x0=[1,1];

x=x0;

dd=subs(df,[x1,x2],x);

while ((norm(dd)>e)&(k<100))

    dm=jacobian(df);

    dm=subs(dm,[x1,x2],x);

    dk=-dm\dd';

    x=nyiwei(x,dk,m);

    dd=subs(df,[x1,x2],x);

    k=k+1;

    if((m*(1/(x(1)+x(2)-1))>e)&(t<100))

    t=t+1;

    m=u*m;

    f=x1^3+x2^2+m*(1/(x1+x2-1));

    df=[diff(f,x1) diff(f,x2) ];

    dd=subs(df,[x1,x2],x);

    else break

    end

end

min=x(1)^3+x(2)^2;

 

%内点法一维搜素

function min=nyiwei(x,dk,m)
syms  x1 x2 ;

f=x1^3+x2^2+m*(1/(x1+x2-1));

a=x;

b=x+dk';

e=0.01;

while(norm(b-a)>e)

    q1=a+0.328*(b-a);

    q2=a+0.618*(b-a);

    f1=subs(f,[x1,x2],q1);

    f2=subs(f,[x1,x2],q2);

    if f1>f2

        a=q1;

    else

        b=q2;

    end

    f=x1^3+x2^2+m*(1/(x1+x2-1));

end

min=(a+b)/2;

外点法和外点法一维搜索

%外点法

function min=waidianfa()
e=0.005;

k=0;

t=0;

m=0.5;

r=2;

syms  x1 x2 ;

f=x1^3+x2^2+m*(x1+x2-1)^2;

df=[diff(f,x1) diff(f,x2) ];

x0=[1,1];

x=x0;

dd=subs(df,[x1,x2],x);

while((norm(dd)>e)&(k<100))

    dm1=[diff(df(1),x1) ,diff(df(1),x2)]';

    dm2=[diff(df(2),x1) ,diff(df(2),x2)]';

    dm=[dm1 ,dm2]';

    dm=subs(dm,[x1,x2],x);

    dk=-dm\dd';

    dk=subs(dk);

    x=wyiwei(x,dk,m);

    k=k+1;

    if (((x(1)+x(2)-1)^2>e)&(t<100))

        t=t+1;

        m=r*m;

        f=x1^3+x2^2+m*(x1+x2-1)^2;

        df=[diff(f,x1) diff(f,x2) ];

        dd=subs(df,[x1,x2],x);

    else break

    end

end

min=x(1)^3+x(2)^2;

disp('采用外点法求解函数:Min{f=x1^3+x2^2}在约束条件{x1+x2-1>=0}的条件下的最优解：')

disp('选取初始点为：'),disp(x0)

disp('迭代次数：'),disp(k)

disp('最小点：'),disp(x)

disp('此时的最小值为'),disp(min)


%外点法一维搜索
function min=wyiwei(x,dk,m)
syms  x1 x2 ;

f=x1^3+x2^2+m*(x1+x2-1)^2;

a=x;

b=x+dk';

e=0.01;

while(norm(b-a)>e)

    q1=a+0.328*(b-a);

    q2=a+0.618*(b-a);

    f1=subs(f,[x1,x2],q1);

    f2=subs(f,[x1,x2],q2);

    if(f1>f2)

        a=q1;

    else

        b=q2;

    end

    f=x1^3+x2^2+m*(x1+x2-1)^2;

end

min=(a+b)/2;