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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 y2y1 h = -h;alpha3 = alpha1;y3 = y1;t = 1;endwhile(1) if t =
3、1 alpha1 = alpha2;y1 = y2; alpha2 = alpha3;y2 = y3; else t = 1; end alpha3 = alpha2+h;y3 = y(x11,x22,d11,d22,alpha3); if y3alpha3 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 alpha = 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*
4、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); y2 = y(x11,x22,d11,d22,alpha2); else b = alpha2;alpha2 =
5、 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 =
6、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(i+1)=f(x(2*i+1),x(2*i+2); end for i=1:n Delta(i)=ff(i)-ff(i+1); end delt
7、a=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); k=k+1; R(k,:)=k,x,d,ff; if f3f0 & (f0-2*f2+f3)*(f0-f2-delta)20.5*delt
8、a*(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*n+2); for i=m:n d(2*i-1)=d(2*i+1); d(2*i)=d(2*i+2); end else if f2f3 x(1)=x(2*n+1); x(2)=x(2*n+2);
9、 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) f = (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fminsear
10、ch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*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 = f
11、loor(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 = fminsearch(f,1,1)x = 4.0000 2.0000fval = -8.0000通过fminsearch函数验算，确认极小值点为(4,2)，极小值为-8。4. 结论函数的极小值点为(4,2)，极小值为-8。5. 程序流程图黄金分割法程序框图开 始给定a、b、c是否是结 束否POWELL算法程序框图开 始是否判别条件是否满足是否是否是结 束否
