程序设之优化设计 MATLAB

Question

大家好！想请教个题目：
题目是：编写一段程序，实现某一优化设计算法，编程语言MATLAB
算出min F（x）=1.5X1^2+0.5X2^2+-X1X2-2*X1
初始点 X0={-2,4} 精度 e=0.01
谢谢 有高分送哦！

Answer 1

最数下降法解无约束优化的程序

先建立一维搜索的m文件：minWP.m如下
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [x,minf]=minWP(f,XMAX,c1,c2,alpha,tol)
%一维搜索的Wolfe-Powell法
%作者:龚纯 王正林<<精通 Matlab 最优化计算>>
%f:目标函数
%XMAX:搜索最大值
%c1:可接受系数1
%c2:可接受系数2
%alpha:增大步长倍数
%tol:精度
%x:极小值点
%minf:极小值点处的函数值

format long;
if nargin==5
tol=1.0e-6;
end

if ~(c1>0)||~(c1<c2)||~(c2<1)||~(XMAX>0)||~(alpha>1)
error('参数不对');
end

var=findsym(f);
df=diff(f);
f0=subs(f,var,0);
df0=subs(df,var,0);
a=0;
b=XMAX;
if b<inf
t=(a+b)/2;
else
t=10;
end

while 1
ft=subs(f,var,t);
fu=f0+c1*t*df0;
if ft<=fu
dft=subs(df,var,t);
dfl=c2*t*df0;
if dft>dfl
x=t;
break;
else
a=t;
if b==inf
t=t*alpha;
else
t=(a+b)/2;
end
continue;
end
else
b=t;
t=(a+b)/2;
continue;
end
end

minf=subs(f,var,t);
format short;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

然后建立最速下降法的m文件如下：

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%minFD.m
function [x,minf]=minFD(f,x0,var,tol)
%最速下降法
%作者：龚纯 王正林《精通Matlab最优化计算》
%目标函数：f
%初始点：x0
%自变量向量：var
%精度：tol
%所求的驻点：x
%驻点处的函数值

format long;
if nargin==3
tol=1.0e-6;
end

gradf=jacobian(f,var); %f的梯度
wucha=1;
syms lamda;

while wucha>tol
d=-subs(gradf,var,x0);
wucha=norm(d);
y=x0+lamda*d;
yf=subs(f,var,y);
%l=minHJ(yf,0,100);
l=minWP(yf,10,0.2,0.6,2);
x1=x0+l*d;
x0=x1;
end

x=x1;
minf=subs(f,var,x);
format short;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
保存后就可调用了：
>> clear
>> syms x1 x2
>> f=1.5*x1^2+0.5*x2^2-x1*x2-2*x1

f =

3/2*x1^2+1/2*x2^2-x1*x2-2*x1

>> [x,mf]=minFD(f,[-2 4],[x1,x2])

x =

1.0000 1.0000

mf =

-1.0000

Answer 2

最数下降法解无约束优化的程序
先建立一维搜索的m文件：minWP.m如下
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function
[x,minf]=minWP(f,XMAX,c1,c2,alpha,tol)
%一维搜索的Wolfe-Powell法
%作者:龚纯
王正林<<精通
Matlab
最优化计算>>
%f:目标函数
%XMAX:搜索最大值
%c1:可接受系数1
%c2:可接受系数2
%alpha:增大步长倍数
%tol:精度
%x:极小值点
%minf:极小值点处的函数值
format
long;
if
nargin==5
tol=1.0e-6;
end
if
~(c1>0)||~(c1<c2)||~(c2<1)||~(XMAX>0)||~(alpha>1)
error('参数不对');
end
var=findsym(f);
df=diff(f);
f0=subs(f,var,0);
df0=subs(df,var,0);
a=0;
b=XMAX;
if
b<inf
t=(a+b)/2;
else
t=10;
end
while
1
ft=subs(f,var,t);
fu=f0+c1*t*df0;
if
ft<=fu
dft=subs(df,var,t);
dfl=c2*t*df0;
if
dft>dfl
x=t;
break;
else
a=t;
if
b==inf
t=t*alpha;
else
t=(a+b)/2;
end
continue;
end
else
b=t;
t=(a+b)/2;
continue;
end
end
minf=subs(f,var,t);
format
short;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
然后建立最速下降法的m文件如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%minFD.m
function
[x,minf]=minFD(f,x0,var,tol)
%最速下降法
%作者：龚纯
王正林《精通Matlab最优化计算》
%目标函数：f
%初始点：x0
%自变量向量：var
%精度：tol
%所求的驻点：x
%驻点处的函数值
format
long;
if
nargin==3
tol=1.0e-6;
end
gradf=jacobian(f,var);
%f的梯度
wucha=1;
syms
lamda;
while
wucha>tol
d=-subs(gradf,var,x0);
wucha=norm(d);
y=x0+lamda*d;
yf=subs(f,var,y);
%l=minHJ(yf,0,100);
l=minWP(yf,10,0.2,0.6,2);
x1=x0+l*d;
x0=x1;
end
x=x1;
minf=subs(f,var,x);
format
short;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
保存后就可调用了：
>>
clear
>>
syms
x1
x2
>>
f=1.5*x1^2+0.5*x2^2-x1*x2-2*x1
f
=
3/2*x1^2+1/2*x2^2-x1*x2-2*x1
>>
[x,mf]=minFD(f,[-2
4],[x1,x2])
x
=
1.0000
1.0000
mf
=
-1.0000

Answer 3

最数下降法解无约束优化的程序
先建立一维搜索的m文件：minWP.m如下
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function
[x,minf]=minWP(f,XMAX,c1,c2,alpha,tol)
%一维搜索的Wolfe-Powell法
%作者:龚纯
王正林<<精通
Matlab
最优化计算>>
%f:目标函数
%XMAX:搜索最大值
%c1:可接受系数1
%c2:可接受系数2
%alpha:增大步长倍数
%tol:精度
%x:极小值点
%minf:极小值点处的函数值
format
long;
if
nargin==5
tol=1.0e-6;
end
if
~(c1>0)||~(c1<c2)||~(c2<1)||~(XMAX>0)||~(alpha>1)
error('参数不对');
end
var=findsym(f);
df=diff(f);
f0=subs(f,var,0);
df0=subs(df,var,0);
a=0;
b=XMAX;
if
b<inf
t=(a+b)/2;
else
t=10;
end
while
1
ft=subs(f,var,t);
fu=f0+c1*t*df0;
if
ft<=fu
dft=subs(df,var,t);
dfl=c2*t*df0;
if
dft>dfl
x=t;
break;
else
a=t;
if
b==inf
t=t*alpha;
else
t=(a+b)/2;
end
continue;
end
else
b=t;
t=(a+b)/2;
continue;
end
end
minf=subs(f,var,t);
format
short;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
然后建立最速下降法的m文件如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%minFD.m
function
[x,minf]=minFD(f,x0,var,tol)
%最速下降法
%作者：龚纯
王正林《精通Matlab最优化计算》
%目标函数：f
%初始点：x0
%自变量向量：var
%精度：tol
%所求的驻点：x
%驻点处的函数值
format
long;
if
nargin==3
tol=1.0e-6;
end
gradf=jacobian(f,var);
%f的梯度
wucha=1;
syms
lamda;
while
wucha>tol
d=-subs(gradf,var,x0);
wucha=norm(d);
y=x0+lamda*d;
yf=subs(f,var,y);
%l=minHJ(yf,0,100);
l=minWP(yf,10,0.2,0.6,2);
x1=x0+l*d;
x0=x1;
end
x=x1;
minf=subs(f,var,x);
format
short;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
保存后就可调用了：
>>
clear
>>
syms
x1
x2
>>
f=1.5*x1^2+0.5*x2^2-x1*x2-2*x1
f
=
3/2*x1^2+1/2*x2^2-x1*x2-2*x1
>>
[x,mf]=minFD(f,[-2
4],[x1,x2])
x
=
1.0000
1.0000
mf
=
-1.0000