优化方法上机大作业学院：姓名：学号：指导老师：肖现涛第一题源程序如下：function zy_x = di1ti(x)%di1ti是用来求解优化作业第一题的函数。

x0=x; yimuxulong=0.000001;g0=g(x0);s0=-g0;A=2*ones(100,100);k=0;while k<100lanmed=-(g0)'*s0/(s0'*A*s0);x=x0+lanmed*s0;g=g(x);k=k+1;if norm(g)<yimuxulongzy_x=x;fprintf('After %d iterations,obtain the optimal solution.\n \n The optimal solution is \n %f.\n\nThe optimal "x" is "ans".',k,f(x) )break;endmiu=norm(g)^2/norm(g0)^2;s=-g+miu*s0;g0=g; s0=s;x0=x;endfunction f=f(x)f=(x'*ones(100,1))^2-x'*ones(100,1);function g=g(x)g=(2*x'*ones(100,1))*ones(100,1)-ones(100,1);代入x0,运行结果如下：>> x=zeros(100,1);>> di1ti(x)After 1 iterations,obtain the optimal solution.The optimal solution is-0.250000.The optimal "x" is "ans".ans =0.005*ones(100,1).第二题1.最速下降法。

源程序如下：function zy_x=di2titidu(x)%该函数用来解大作业第二题。

x0=x; yimuxulong=1e-5; k=0;g0=g(x0); s0=-g0;while k>=0if norm(g0)<yimuxulongbreak;elselanmed=10;c=0.1;i=0;while i>=0&i<100x=x0+lanmed*s0;if f(x)>(f(x0)+c*lanmed*g0'*s0) lanmed=lanmed/2;i=i+1;elsebreak;endendx=x0+lanmed*s0;x0=x;g0=g(x);s0=-g0;k=k+1;endendzy_x=x;zyj=f(x);fprintf('after %d iterations,obtain the optimal solution.\n\nThe optimal solution is %f.\n\n The optimal "x" is "ans".\n',k,zyj);function f=f(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;f=(x1-1)^2+(x3-1)^2+100*(x2-x1^2)^2+100*(x4-x3^2)^2;function g=g(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;g=[2*(x1-1)-400*x1*(x2-x1^2);200*(x2-x1^2);2*(x3-1)-400*x3*(x4-x3^2);200*(x 4-x3^2)];>> x=[-1.2 1 -1.2 1]';>> di2titidu(x)after 5945 iterations,obtain the optimal solution. The optimal solution is 0.000000.The optimal "x" is "ans".ans =1.00001.00001.00001.00002.牛顿法源程序如下：function zy_x=di2tinewton(x)%该函数用来解大作业第二题。

x0=x; yimuxulong=1e-5; k=0;g0=g(x0); h0=h(x0);s0=-inv(h0)*g0;while k>=0&k<1000if norm(g0)<yimuxulongbreak;elsex=x0+s0;x0=x;g0=g(x);h0=h(x);s0=-inv(h0)*g0;k=k+1;endendzy_x=x;zyj=f(x);fprintf('after %d iterations,obtain the optimal solution.\n\nThe optimal solution is %f.\n\n The optimal "x" is "ans".\n',k,zyj);function f=f(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;f=(x1-1)^2+(x3-1)^2+100*(x2-x1^2)^2+100*(x4-x3^2)^2;function g=g(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;g=[2*(x1-1)-400*x1*(x2-x1^2);200*(x2-x1^2);2*(x3-1)-400*x3*(x4-x3^2);200*(x 4-x3^2)];function h=h(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;h=[2+1200*x1^2-400*x2 -400*x1 0 0;-400*x1 200 0 0;0 0 2+1200*x3^2-400*x4 -400*x3;0 0 -400*x3 200];代入初始值，运行结果如下：>> x=[-1.2 1 -1.2 1]';>> di2tinewton(x)after 6 iterations,obtain the optimal solution.The optimal solution is 0.000000.The optimal "x" is "ans".ans =1.00001.00001.00001.0000可以看出，用Newton法经过6次迭代就能求出最优解。

3.BFGS法源程序如下：function zy_x=di2tiBFGS(x)%该函数用来解大作业第二题。

x0=x; yimuxulong=1e-5; k=0;g0=g(x0); H0=eye(4);s0=-H0*g0;while k>=0&k<100if norm(g0)<yimuxulongbreak;elselanmed=10;c=0.1;i=0;while i>=0&i<100x=x0+lanmed*s0;if f(x)>(f(x0)+c*lanmed*g0'*s0)lanmed=lanmed/2;i=i+1;elsebreak;endendx=x0+lanmed*s0;dete_x=x-x0;dete_g=g(x)-g0;miu=1+dete_g'*H0*dete_g/(dete_x'*dete_g);H=H0+(miu*dete_x*dete_x'-H0*dete_g*dete_x'-dete_x*dete_g'*H0)/(dete_x'*dete _g);s=-H*g(x);x0=x;s0=s;H0=H;g0=g(x);k=k+1;endendzy_x=x;zyj=f(x);fprintf('after %d iterations,obtain the optimal solution.\n\nThe optimal solution is %f.\n\n The optimal "x" is "ans".\n',k,zyj);function f=f(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;f=(x1-1)^2+(x3-1)^2+100*(x2-x1^2)^2+100*(x4-x3^2)^2;function g=g(x)x1=[1 0 0 0]*x;x2=[0 1 0 0]*x;x3=[0 0 1 0]*x;x4=[0 0 0 1]*x;g=[2*(x1-1)-400*x1*(x2-x1^2);200*(x2-x1^2);2*(x3-1)-400*x3*(x4-x3^2);200*(x 4-x3^2)];代入初始值，计算结果如下：>> x=[-1.2 1 -1.2 1]';>> di2tiBFGS(x)after 53 iterations,obtain the optimal solution.The optimal solution is 0.000000.The optimal "x" is "ans".ans =1.00001.00001.00001.0000第三题1.惩罚函数法源程序如下：function zy_x=di3ti(x)%该函数用来解大作业第三题。

x0=x; M=100; c=4; m=1;while m>0g0=g(x0,M); yimuxulong=1e-5;k=0;s0=-inv(H(x0,M))*g0; while k>=0if norm(g0)<yimuxulongbreak;elsex=x0+s0; %牛顿法；x0=x;g0=g(x,M);s0=-inv(H(x0,M))*g0;k=k+1;endendif max([abs(h(x)),g1(x),g2(x),g3(x)])<0.5break;elseM=M*c;m=m+1;endendzy_x=x;zyj=f(x);fprintf('after %d iterations,obtain the optimal solution.\n\nThe optimal solution is %f.\n\n The optimal "x" is "ans".\n',m,zyj);function F=F(x,M)x1=[1 0]*x;x2=[0 1]*x;F=4*x1-x2^2-12+M*(h^2+g1^2+g2^2+g3^2);function g=g(x,M)x1=[1 0]*x;x2=[0 1]*x;g=[4+M*(-4*(25-x1^2-x2^2)*x1+2*(10*x1-x1^2+10*x2-x2^2-34)*(10-2*x1)+2*x1);-2*x2+M*(-4*(25-x1^2-x2^2)*x2+2*(10*x1-x1^2+10*x2-x2^2-34)*(10-2*x2)+2*x2)];function H=H(x,M)x1=[1 0]*x;x2=[0 1]*x;H=[M*(24*x1^2-120*x1+8*x2^2-40*x2+238),M*(16*x1*x2-40*x1-40*x2+200);M*(16*x 1*x2-40*x1-40*x2+200),-2+M*(24*x2^2-120*x2+8*x1^2-40*x1+238)];function f=f(x)x1=[1 0]*x;x2=[0 1]*x;f=4*x1-x2^2-12;function h=h(x)x1=[1 0]*x;x2=[0 1]*x;h=25-x1^2-x2^2;function g1=g1(x)x1=[1 0]*x;x2=[0 1]*x;g=10*x1-x1^2+10*x2-x2^2-34; if g<0g1=g;elseg1=0;endfunction g2=g2(x)x1=[1 0]*x;x2=[0 1]*x;if x1>=0g2=0;elseg2=x1;endfunction g3=g3(x)x1=[1 0]*x;x2=[0 1]*x;if x2>=0g3=0;elseg3=x2;end代入任意初始值，运算结果如下。