function maximization and different parameter values

Hi

I'm a matlab beginner.

I'd like to try maximization in matlab.

This is the function: f=(x(1)^(1/m)+x(2)^(1/m))^m.

where m is a parameter,

and x(1)+x(2)=1, x(1)>=0, x(2)>=0.

I tried the following code, but it's not working.

if true
  % code 
clc;
f = @(x,m)(-(x(1)^(1/m)+x(2)^(1/m))^m)
m=1:0.1:1.4
for ii = 1:length(m)
[x(:,ii),fval]=fmincon(@(x)f(x,m(ii)),x0,[1,1],1,[],[],[0,0],[inf,inf]) 
plot(x(1,:)-x(2,:),f)

end

I want to show that only when x(1)=x(2), f is maximized,

and want to draw a single plot for functions with different parameter values like 1, 1.1, 1.2....

Btw the plot... its horizontal axis of the plot should be x(1)-x(2), and the vertical axis is f.

Can anyone please help me solve this problem?

Thank you in advance!

Answer 1

[Matt J]

Did you really mean the constraint x(1)+x(2)=1 to be an equality instead of an inequality? If so, you are calling fmincon with the wrong syntax. Also, your function behaves badly/non-differentiably for negative x and m>1, so you have to make sure the algorithm iterations never go there. Using the sqp algorithm and forcing f to return NaN for non-positive x(i) can help with this,

opts=optimoptions('Algorithm','sqp');

f=@(x,m)(-(x(1)^(1/m)+x(2)^(1/m))^m)/all(x>0);

   ...

[x(:,ii),fval]=fmincon(@(x)f(x,m(ii)),x0,[],[],[1,1],1,[0,0],[],[],opts); 

   ...

[Lena]

Dear Matt,

Thanks for the reply.

I tried it again with your suggestion, but I got this message: Undefined function or method 'optimoptions' for input arguments of type 'char'...

Could this be due to different versions of Matlab perhaps? I'm lost..

the given condition is m>1..and yes x(1)+x(2)=1 equality seems to be the constraint here.

In fact, I'm trying to reproduce a graph I saw somewhere. The plot was a set of concave functions with each different parameter, where their maximum points were achieved at x(1)-x(2)=0. And the higher the m, the higher output values. The horizontal axis is x(1)-x(2).

Please any advice will help me a lot. :)

[Matt J]

Could this be due to different versions of Matlab perhaps? I'm lost..

Yes, use optimset instead. What MATLAB version do you have? You may also have to use the interior-point algorithm instead of sqp.

The horizontal axis is x(1)-x(2).

But at the solution x(1)-x(2)=0 for all m. You are trying to plot a vertical line?

[Lena]

I will try again with your suggestion, and will try the previous code in other versions. I'm using 2011.

Yes. I'd like to plot of output f versus (x(1)-x(2)) for each different values of m. How can I draw this? ..

[Lena]

I tried again and got the result! Thank you!

But I still cannot draw the plot I want... f as the vertical axis and x(1)-x(2) as the horizontal axis, I should get a set of concave functions where the max. is at 0=x(1)-x(2).

[Matt J]

No, the function f is not purely a function of x(1)-x(2). For example, with m=1, when x=[1,0], the value of f is 1. However, when x=[2,1], the value of f is 3. Thus, you get different values for f even though x(2)-x(1)=1 and m is the same in both cases.

[Lena]

yes, with m=1, the curve would be just the horizontal line on the axis..so many different values for m=1. but with other m's >1, there should be concave curves. but i don't know how to replicate the graph.. nyway, thanks a lot, matt! your answer really helped me a lot!

[Matt J]

but with other m's >1, there should be concave curves

Nope. My remarks applied to general m. Your objective cannot be written as a function of x(1)-x(2) for any m.

nyway, thanks a lot, matt! your answer really helped me a lot!

You're welcome!