优化方法作业第二版..doc
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优化方法上机大作业
院系:化工与环境生命学部
姓名:李翔宇
学号:31607007
指导教师:肖现涛
第一题:
1.最速下降法
源程序如下:
function x_star = ZSXJ(x0,eps)
gk = grad(x0);
res = norm(gk);
k = 0;
while res > eps && k<=10000
dk = -gk;
ak =1; f0 = fun(x0);
f1 = fun(x0+ak*dk);
slope = dot(gk,dk);
while f1 > f0 + 0.0001*ak*slope
ak = ak/2;
xk = x0 + ak*dk;
f1 = fun(xk);
end
k = k+1;
x0 = xk;
gk = grad(xk);
res = norm(gk);
fprintf('--The %d-th iter, the residual is %f\n',k,res);
end
x_star = xk;
end
function f = fun(x)
f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2;
end
function g = grad(x)
g = zeros(2,1);
g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);
end
运行结果:
>> x0=[0,0]';
>> esp=1e-4;
>> xk=ZSXJ(x0,eps)
--The 1-th iter, the residual is 13.372079 --The 2-th iter, the residual is 12.079876 --The 3-th iter, the residual is 11.054105 ……………………………………………
--The 9144-th iter, the residual is 0.000105 --The 9145-th iter, the residual is 0.000102 --The 9146-th iter, the residual is 0.000100 xk =
0.9999
0.9998
MATLAB截屏:
2.牛顿法
源程序如下:
function x_star = NEWTON(x0,eps)
gk = grad(x0);
bk = [grad2(x0)]^(-1);
res = norm(gk);
k = 0;
while res > eps && k<=1000
dk=-bk*gk;
xk=x0+dk;
k = k+1;
x0 = xk;
gk = grad(xk);
bk = [grad2(xk)]^(-1);
res = norm(gk);
fprintf('--The %d-th iter, the residual is %f\n',k,res); end
x_star = xk;
end
function f = fun(x)
f = (1-x(1))^2 + 100*(x(2)-x(1)^2)^2;
end
function g = grad2(x)
g = zeros(2,2);
g(1,1)=2+400*(3*x(1)^2-x(2));
g(1,2)=-400*x(1);
g(2,1)=-400*x(1);
g(2,2)=200;
end
function g = grad(x)
g = zeros(2,1);
g(1)=2*(x(1)-1)+400*x(1)*(x(1)^2-x(2)); g(2) = 200*(x(2)-x(1)^2);
end
运行结果:
>> x0=[0,0]';eps=1e-4;
>> xk=NEWTON(x0,eps)
--The 1-th iter, the residual is 447.213595 --The 2-th iter, the residual is 0.000000 xk =
1.0000
1.0000
MATALB截屏;
3.BFGS方法
源程序如下:
function x_star = Bfgs(x0,eps)
g0 = grad(x0);
gk=g0;
res = norm(gk);
Hk=eye(2);
k = 0;
while res > eps && k<=1000
dk = -Hk*gk;
ak =1; f0 = fun(x0);
f1 = fun(x0+ak*dk);
slope = dot(gk,dk);
while f1 > f0 + 0.1*ak*slope
ak = ak/2;
xk = x0 + ak*dk;
f1 = fun(xk);
end
k = k+1;
fa0=xk-x0;
x0 = xk;
g0=gk;
gk = grad(xk);
y0=gk-g0;
Hk=((eye(2)-fa0*(y0)')/((fa0)'*(y0)))*((eye(2)-
(y0)*(fa0)')/((fa0)'*(y0)))+(fa0*(fa0)')/((fa0)'*(y0)); res = norm(gk);
fprintf('--The %d-th iter, the residual is %f\n',k,res); end
x_star = xk;
end
function f=fun(x)