用Powell法编程求解无约束优化问题

用Powell法编程求解无约束优化问题题目：用Powell法编程求解无约束优化问题min，要求：（1）程序流程图（2）源程序清单（3）运行结果求解过程：流程图....程序清单源程序由以下几个部分组成：（1）目标函数程序清单：建立函数f（）以f.m存盘%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%函数名：f（），参数2个%函数作用：带入参数，求解目标函数f（）%可以更改目标函数为任意二维函数，结果由m返回%作者：WYH学号：xxxxxxxx%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionm=f(x1,x2)m=x1^2+x2^2-x1*x2-10*x1-4*x2+60;%目标函数（2）关于α的目标函数清单：建立函数y（）以y.m存盘%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%函数名：y（），参数5个%函数作用：带入参数，求解关于α的一元函数y（α），根据原目标函数更改此函数，%其中原目标函数变量X1＝x1+alpha*d1，X2＝x2+alpha*d2结果由m返回%作者：WYH学号：xxxxxxxx%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionm=y(x1,x2,d1,d2,alpha)m=(x1+alpha*d1)^2+(x2+alpha*d2)^2-(x1+alpha*d1)*(x2+alpha*d2)-10*(x1+alpha*d1)-4*(x2+alpha*d2)+60;%含α的目标函数，其形式与原目标函数f（x）＝x1^2+x2^2-x1*x2-10*x1-4*x2+60一致（3）外推法求解一元函数最小值区间：建立函数section（）以sextion.m存盘%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%函数名：section（），参数4个%函数作用：利用外推法求解关于α的函数y（α）的极小点α*所在的区间【a，b】，%结果由a、b返回%作者：WYH学号：xxxxxxxx%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function[a,b]=section(x1,x2,d1,d2)x11=x1;%给出起始点坐标x1，x2和搜索方向d1，d2x22=x2;d11=d1;d22=d2;h0=1;%初始化h=h0;alpha1=0;y1=y(x11,x22,d11,d22,alpha1);%代入α1求解y1alpha2=h;y2=y(x11,x22,d11,d22,alpha2);t=0;ify2y1h=-h;alpha3=alpha1;y3=y1;t=1;%如果y2y1，则改变搜索方向endwhile(1)ift==1alpha1=alpha2;y1=y2;%实现交换alpha2=alpha3;y2=y3;elset=1;endalpha3=alpha2+h;y3=y(x11,x22,d11,d22,alpha3);ify3y2h=2*h;%改变搜索步长，将其加倍elsebreak;endendifalpha1alpha3tem=alpha1;alpha1=alpha3;alpha3=tem;%比较大小，保证输出区间为【a，b】a=alpha1;b=alpha3;elsea=alpha1;b=alpha3;end（1）黄金分割法求解区间已知的一元函数最小值：建立函数ALPHA（）以ALPHA.m存盘%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%函数名：ALPHA（），参数6个%函数作用：利用黄金分割法求解关于α的函数y（α）的极小点α*，已知α所在的%区间【A，B】，结果由α返回%作者：WYH学号：xxxxxxxx%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionalpha=ALPHA(x1,x2,d1,d2,A,B)x11=x1;x22=x2;%给出起始点坐标x1，x2和搜索方向d1，d2d11=d1;d22=d2;a=A;b=B;%获取区间epsilon=0.000001;%初始化，给定进度r=0.618;alpha1=b-r*(b-a);y1=y(x11,x22,d11,d22,alpha1);%代入α1求解y1alpha2=a+r*(b-a);y2=y(x11,x22,d11,d22,alpha2);while(1)ify1=y2%根据区间消去法原理缩短搜索空间a=alpha1;alpha1=alpha2;y1=y2;alpha2=a+r*(b-a);y2=y(x11,x22,d11,d22,alpha2);elseb=alpha2;alpha2=alpha1;y2=y1;alpha1=b-r*(b-a);y1=y(x11,x22,d11,d22,alpha1);endifabs(b-a)epsilon&abs(y2-y1)epsilon%判断是否满足进度要求break;%满足进度要求则退出循环迭代过程endendalpha=0.5*(a+b);%返回值（2）主程序，建立Powell.m存盘，运行程序%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%程序名：powell法求解二维函数的极小值，主程序%程序作用：根据无约束优化方法中的Powell法原理，在二维空间求解f（x1，x2）的%最小点，结果返回最小点变量坐标（x1，x2）及最小值fmin%作者：WYH学号：xxxxxxxx%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clearallclck=0;n=2;x=[0;0;];ff(1)=f(x(1),x(2));%初始化epsilon=0.00001;d=[1;0;0;1;];while(1)x00=[x(1);x(2);];fori=1:n[a(i),b(i)]=section(x(2*i-1),x(2*i),d(2*i-1),d(2*i));%调用section（）函数求解一元函数有y（α）最小值时的区间alpha(i)=ALPHA(x(2*i-1),x(2*i),d(2*i-1),d(2*i),a(i),b(i));%调用ALPHA（）函数求解y（α）最小值点α*x(2*i+1)=x(2*i-1)+alpha(i)*d(2*i-1);%沿di方向搜索x(2*i+2)=x(2*i)+alpha(i)*d(2*i);ff(i+1)=f(x(2*i+1),x(2*i+2));%搜索到的点对应的函数值endfori=1:nDelta(i)=ff(i)-ff(i+1);enddelta=max(Delta);%求出函数值之差的最大值fori=1:n%寻找函数值之差最大值的下标ifdelta==Delta(i)m=i;break;endendd(2*n+1)=x(2*n+1)-x(1);%求出反射点搜索方向dn+1d(2*n+2)=x(2*n+2)-x(2);x(2*n+3)=2*x(2*n+1)-x(1);%搜索到反射点Xn+1x(2*n+4)=2*x(2*n+2)-x(2);ff(n+2)=f(x(2*n+3),x(2*n+4));%反射点所对应的函数值f0=ff(1);f2=ff(n+1);f3=ff(n+2);k=k+1;%记录迭代次数R(k,:)=[k,x',d',ff];%保存迭代过程的中间运算结果iff3f0&(f0-2*f2+f3)*(f0-f2-delta)^20.5*delta*(f0-f3)^2%判断是否需要对原方向组进行替换[a(n+1),b(n+1)]=section(x(2*n+1),x(2*n+2),d(2*n+1),d(2*n+2));alpha(n+1)=ALPHA(x(2*n+1),x(2*n+2),d(2*n+1),d(2*n+2),a(n+1),b(n+1));x(1)=x(2*n+1)+alpha(n+1)*d(2*n+1);%沿反射方向进行搜索，将搜索结果作为下一轮迭代的始点x(2)=x(2*n+2)+alpha(n+1)*d(2*n+2);fori=m:n%根据函数值之差最大值的下标值m，对原方向组进行替换d(2*i-1)=d(2*i+1);d(2*i)=d(2*i+2);endelseiff2f3%如果不需要对原方向组进行替换，选取终点及反射点中函数值较小者作为下一轮迭代的始点x(1)=x(2*n+1);x(2)=x(2*n+2);elsex(1)=x(2*n+3);x(2)=x(2*n+4);endendRR(k,:)=alpha;%保存迭代过程的中间运算结果ff(1)=f(x(1),x(2));%计算下一轮迭代过程需要的f0值if(((x(2*n+1)-x00(1))^2+(x(2*n+2)-x00(2))^2)^(1/2))epsilon%判断是否满足精度要求break;%满足进度要求则退出循环迭代过程endendxx=[x(1);x(2)]fmin=f(x(1),x(2))%显示最小值及其所对应的坐标二、运行结果xx=8.00006.0000fmin=8.0000