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@@ -1,85 +1,85 @@ |
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syms x1 x2;
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X = [x1;x2];
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fx = 2*x1^2 + 2*x1*x2 + 2*x2^2 - 4*x1 - 6*x2;
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x=[1;1];
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e=0.0001;
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minVal = newton(fx,X,x,e);
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function [ y ] = newton(fx,var,x,e,MAX)
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% 牛顿法求解函数极小点
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% 参数描述------------------------------
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% fx 符号表达式
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% var 符号变量列表 如：syms x1 x2;var= [x1;x2];
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% x 起始位置
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% e 精度控制
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% MAX 最大迭代次数控制
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% ------------------------------
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% Armijo步长
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step=2;% 初始步长设定
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r=0.5; % 取值范围(0,1),越大越快
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b=0.5; % 取值范围(0,1),越大越慢
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if nargin < 5
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MAX = 100;%设置默认最大迭代次数
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end
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precision = 5;%显示精度控制
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n=length(x);
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%开始循环迭代
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%direction存贮搜索方向
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%step 存贮步长
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H=eye(n);
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bfound = 0;
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for k=1:1:MAX
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[gx,direction] = getNextDirecrion(fx,var,x,H);
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disp('------------------------------');
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% fprintf('d[%d]=:',k);
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% disp( vpa(direction',precision) );
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if abs(subs(gx,var,x)) <= e
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y = x;
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bfound = 1;%得到结果时置为1
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break;
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else
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x0 = x+step*direction;
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while subs(fx,var,x0) > subs(fx,var,x) + b*step*r*subs(gx',var,x)*direction
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step = step*r;
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x0 = x+step*direction;
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end
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%打印求解过程
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fprintf('X[%d]=:',k);
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disp( vpa(x',precision) );
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fprintf('step(%d)=: ', k);
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disp( vpa(step,precision) );
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disp('------------------------------');
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x1 = x;
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x = x+step*direction;%计算下一个位置
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end
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H=getH(fx,var,x1,x,H);
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end
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if bfound == 1
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fprintf('X[%d]=:',k);
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disp( vpa(x',precision) );
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fprintf('step(%d)=: ', k);
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disp( vpa(step,precision) );
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disp('min value of:');
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fprintf("%.6f\n", vpa( subs(fx,var,y),precision) );
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end
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end
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%根据位置xk,获取搜索方向
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function [gx,direction] = getNextDirecrion(fx,var,xk,H)
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gx = gradient(fx,var); %计算梯度函数
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direction = -H^(-1)*subs(gx,var,xk);%计算搜索方向
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end
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%
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function [H] =getH(fx,var,x0,x1,H)
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gx = gradient(fx,var); %计算梯度函数
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s = x1 - x0;
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y = subs(gx,var,x1) - subs(gx,var,x0);
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H = H + y*y'/(y'*s) - H*s*s'*H/(s'*H*s);%BFGS
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% H = H + (y-H*s)*(y-H*s)'/((y-H*s)'*s);%秩1
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% H = H - H*y*y'*H/(y'*H*y) + s*s'/(s'*y); %DFP
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syms x1 x2; |
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X = [x1;x2]; |
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fx = 2*x1^2 + 2*x1*x2 + 2*x2^2 - 4*x1 - 6*x2; |
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x=[1;1]; |
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e=0.0001; |
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minVal = newton(fx,X,x,e); |
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function [ y ] = newton(fx,var,x,e,MAX) |
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% 牛顿法求解函数极小点 |
|
|
|
% 参数描述------------------------------ |
|
|
|
% fx 符号表达式 |
|
|
|
% var 符号变量列表 如：syms x1 x2;var= [x1;x2]; |
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|
% x 起始位置 |
|
|
|
% e 精度控制 |
|
|
|
% MAX 最大迭代次数控制 |
|
|
|
% ------------------------------ |
|
|
|
|
|
|
|
% Armijo步长 |
|
|
|
step=2;% 初始步长设定 |
|
|
|
r=0.5; % 取值范围(0,1),越大越快 |
|
|
|
b=0.5; % 取值范围(0,1),越大越慢 |
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if nargin < 5 |
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MAX = 100;%设置默认最大迭代次数 |
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end |
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precision = 5;%显示精度控制 |
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n=length(x); |
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%开始循环迭代 |
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%direction存贮搜索方向 |
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%step 存贮步长 |
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H=eye(n); |
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bfound = 0; |
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for k=1:1:MAX |
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[gx,direction] = getNextDirecrion(fx,var,x,H); |
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disp('------------------------------'); |
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% fprintf('d[%d]=:',k); |
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% disp( vpa(direction',precision) ); |
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if abs(subs(gx,var,x)) <= e |
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y = x; |
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bfound = 1;%得到结果时置为1 |
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break; |
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else |
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x0 = x+step*direction; |
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while subs(fx,var,x0) > subs(fx,var,x) + b*step*r*subs(gx',var,x)*direction |
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step = step*r; |
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x0 = x+step*direction; |
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end |
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%打印求解过程 |
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fprintf('X[%d]=:',k); |
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disp( vpa(x',precision) ); |
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fprintf('step(%d)=: ', k); |
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disp( vpa(step,precision) ); |
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disp('------------------------------'); |
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x1 = x; |
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x = x+step*direction;%计算下一个位置 |
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end |
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H=getH(fx,var,x1,x,H); |
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end |
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if bfound == 1 |
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fprintf('X[%d]=:',k); |
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disp( vpa(x',precision) ); |
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fprintf('step(%d)=: ', k); |
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disp( vpa(step,precision) ); |
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disp('min value of:'); |
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fprintf("%.6f\n", vpa( subs(fx,var,y),precision) ); |
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end |
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end |
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%根据位置xk,获取搜索方向 |
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function [gx,direction] = getNextDirecrion(fx,var,xk,H) |
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gx = gradient(fx,var); %计算梯度函数 |
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direction = -H^(-1)*subs(gx,var,xk);%计算搜索方向 |
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end |
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%拟牛顿条件 |
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function [H] =getH(fx,var,x0,x1,H) |
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gx = gradient(fx,var); %计算梯度函数 |
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s = x1 - x0; |
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y = subs(gx,var,x1) - subs(gx,var,x0); |
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H = H + y*y'/(y'*s) - H*s*s'*H/(s'*H*s);%BFGS |
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% H = H + (y-H*s)*(y-H*s)'/((y-H*s)'*s);%秩1 |
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% H = H - H*y*y'*H/(y'*H*y) + s*s'/(s'*y); %DFP |
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end |
