算法原理

to-do

Matlab代码

clc; clear;

f = @(x) x(1).^2+2*x(1)*x(2)+3*x(2).^2; %待求函数，x1,x2,x3...

% f = @(x) x(1).^2+2*x(2).^2;

paraNum = 2; %函数参数的个数，x1,x2,x3...的个数

x0 = [3,3]; %初始值

tol = 1e-5; %迭代容忍度

flag = inf; %结束条件

error = []; %函数变化

while flag > tol

p = g(f,x0,paraNum); %列向量

f2 = @(a) f(x0-a*p');

buChang = argmin(f2); %求步长，line search：argmin function

x1 = x0-buChang*p';

flag = norm(x1-x0);

error = [error,flag];

x0 = x1;

end

plot(0:length(error)-1,error)

function [f_grad] = g(f,x0,paraNum)

temp = sym('x',[1,paraNum]);

f1=f(temp);

Z = gradient(f1);

f_grad = double(subs(Z,temp,x0));

end

function [x] = argmin(f)

%求步长

t = 0;

options = optimset('Display','off');

[x,~] = fminunc(f,t,options);

end

代码问题

Matlab符号运算，耗时

最速下降法的步长使用line-search，耗时

代码改进

clc; clear;

f = @(x) x(1).^2+2*x(1)*x(2)+3*x(2).^2; %待求函数，x1,x2,x3...

% f = @(x) x(1).^2+2*x(2).^2;

paraNum = 2; %函数参数的个数，x1,x2,x3...的个数

x0 = [3,3]; %初始值

tol = 1e-3; %迭代容忍度

flag = inf; %结束条件

error = []; %函数变化

while flag > tol

% for i =1:1

p = g(f,x0,paraNum); %列向量

if norm(p) < tol

buChang = 0;

else

buChang = argmin(f,x0,p,paraNum); %求步长，line search：argmin function

end

x1 = x0-buChang.*p';

flag = norm(x1-x0);

error = [error,flag];

x0 = x1;

end

plot(0:length(error)-1,error)

function [f_grad] = g(f,x0,paraNum)

temp = sym('x',[1,paraNum]);

f1=f(temp);

Z = gradient(f1);

f_grad = double(subs(Z,temp,x0));

end

% function [x] = argmin(f,paraNum)

% %求步长

% t = zeros(1,paraNum);

% options = optimset('Display','off');

% [x,~] = fminunc(f,t,options);

% end

function [x] = argmin(f,x0,p,num)

% 求步长

% for i=1:paraNum

% syms(['x',num2str(i)]);

% end

temp = sym('x',[1,num]);

f1=f(x0 - temp.*p');

for i = 1:num

temp(i) = diff(f1,temp(i));

end

jieGuo = solve(temp);

jieGuo = struct2cell(jieGuo);

x = zeros(1,num);

for i = 1:num

x(i) = double(jieGuo{i});

end

end