用Powell法优化设计程序与一维搜索黄金分割法组合

1、用Powell法优化设计程序与一维搜索黄金分割法组合编程求解函数的极小点x，初始点x0=1,1T，迭代精度=0.001。解：已知，初始点x0=1,1T，迭代精度=0.001。在该优化设计过程中，黄金分割搜索法作为POWELL算法主程序中的一部分。在POWELL算法运行过程中会多次调用黄金分割搜索算法程序。这样可以缩短优化设计计算时间。1. MATLAB源程序代码1.1 关于的目标函数源代码function m = y(x1,x2,d1,d2,alpha)%建立关于的一元函数y()m = (x1+alpha*d1)2+2*(x2+alpha*d2)2-4*(x1+alpha*d1)-2*(x1+

2、alpha*d1)*(x2+alpha*d2);1.2 一元函数最小值区间函数源代码function a,b = section(x1,x2,d1,d2)%采用外推法求解一元函数的最小值区间x11 = x1;x22 = x2;d11 = d1;d22 = d2;h0 = 1;h = h0;alpha1 = 0;y1 = y(x11,x22,d11,d22,alpha1);alpha2 = h;y2 = y (x11,x22,d11,d22,alpha2);t = 0;if y2>y1 h = -h;alpha3 = alpha1;y3 = y1;t = 1;endwhile(1) if

3、t = 1 alpha1 = alpha2;y1 = y2; alpha2 = alpha3;y2 = y3; else t = 1; end alpha3 = alpha2+h;y3 = y(x11,x22,d11,d22,alpha3); if y3<y2 h = 2*h; else break; endendif alpha1>alpha3 tem = alpha1;alpha1 = alpha3;alpha3 = tem; a = alpha1;b = alpha3;else a=alpha1;b = alpha3;end1.3黄金分割搜索法函数源代码function al

4、pha = ALPHA(x1,x2,d1,d2,A,B)%利用黄金分割法求解关于的函数y()的极小点*x11 = x1;x22 = x2;d11 = d1;d22 = d2;a = A;b = B;ep = 0.001;r = 0.618;alpha1 = b-r*(b-a);y1 = y(x11,x22,d11,d22,alpha1);alpha2 = a+r*(b-a);y2 = y(x11,x22,d11,d22,alpha2);while(1) if y1>=y2 a = alpha1;alpha1 = alpha2; y1 = y2; alpha2 = a+r*(b-a); y

5、2 = y(x11,x22,d11,d22,alpha2); else b = alpha2;alpha2 = alpha1; y2 = y1; alpha1 = b-r*(b-a); y1 = y(x11,x22,d11,d22,alpha1); end if abs(b-a)<ep && abs(y2-y1)<ep break; endendalpha = 0.5*(a+b);1.4 POWELL算法程序源代码function z,fmin = powell(f) %根据无约束优化方法POWELL算法，在二维空间中求解f(x1,x2)的最小值点，求解结果返回变量

6、坐标(x1,x2)和极小值fmink = 0;n = 2;x = 0;0;ff(1) = f(x(1),x(2);ep = 0.001;d = 1;0;0;1;while(1) x00 = x(1);x(2); for i = 1:n a(i),b(i)=section(x(2*i-1),x(2*i),d(2*i-1),d(2*i); alpha(i)=ALPHA(x(2*i-1),x(2*i),d(2*i-1),d(2*i),a(i),b(i); x(2*i+1)=x(2*i-1)+alpha(i)*d(2*i-1); x(2*i+2)=x(2*i)+alpha(i)*d(2*i); ff(

7、i+1)=f(x(2*i+1),x(2*i+2); end for i=1:n Delta(i)=ff(i)-ff(i+1); end delta=max(Delta); for i = 1:n if delta = Delta(i) m = i; break; end end d(2*n+1) = x(2*n+1)-x(1); d(2*n+2)=x(2*n+2)-x(2); x(2*n+3)=2*x(2*n+1)-x(1); x(2*n+4)=2*x(2*n+2)-x(2); ff(n+2)=f(x(2*n+3),x(2*n+4); f0=ff(1);f2=ff(n+1);f3=ff(n+2

8、); k=k+1; R(k,:)=k,x',d',ff; if f3<f0 && (f0-2*f2+f3)*(f0-f2-delta)2<0.5*delta*(f0-f3)2 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); x(2)=x(2*n+2)+alpha(n+1)*d(2

9、*n+2); for i=m:n d(2*i-1)=d(2*i+1); d(2*i)=d(2*i+2); end else if f2<f3 x(1)=x(2*n+1); x(2)=x(2*n+2); else x(1)=x(2*n+3); x(2)=x(2*n+4); end end RR(k,:)=alpha; ff(1)=f(x(1),x(2); if (x(2*n+1)-x00(1)2+(x(2*n+2)-x00(2)2)(1/2)<ep break; endendz = x(1);x(2);fmin = f(x(1),x(2);2. 运行程序计算>> f =

10、(x1,x2) x12+2*x22-4*x1-2*x1*x2f = (x1,x2)x12+2*x22-4*x1-2*x1*x2>> z,fmin = powell(f)z = 3.9998 1.9998fmin = -8.0000>>经过计算可知，极小值点为(3.9998,1.9998)，极小值为-8。3. 验算通过MATLAB软件内置fminsearch函数进行验算。3.1 绘图>> x = linspace(-10,10,50);y = linspace(-10,10,50);x,y = meshgrid(x,y);z = x.2+2*y.2-4*x-2

11、*x.*y;subplot(1,2,1);%在一幅框中显示两张图，显示第一张图cs = contour(x,y,z);clabel(cs); %绘制等高线图xlabel('x1');ylabel('x2');title('(a)Contour plot');grid;>> subplot(1,2,2); %在一幅框中显示两张图，显示第二张图cs = surfc(x,y,z); %绘制空间曲面图zmin = floor(min(z);zmax = ceil(max(z);xlabel('x1');ylabel('x2');zlabel('f(x1,x2)');title('(b) Mesh plot');>>通过观察可以发现极小值点在(4,2)附近。3.2 fminsearch函数验算>> f = (x) x(1)2+2*x(2)2-4*x(1)-2*x(1)*x(2)f = (x)x(1)2+2*x(2)2-4*x(1)-2*x(1)*x(2)>> x,fval = fm