How to find max and min value of a function ?

Hi ... I have a function u_g

a=2; b=2; c=1; e=0.75 ; l=0.5;
u_g = @(x, x_e, N)(-0.5*a*x.^2+b*(x-x_e)-c*(N.^l)+e*u_p(x,x_e,N));

x and N are variables, the rest of them are parameters... I want to find the value of x which will bring u_g the maximum value and corresponding max value of u_g

and I want to find the value of N which will bring minimum value to u_g and corresponding value of u_g ..how can I do it ?.. do I have to fix one of the variables?

[Walter Roberson]

u_p is a function?

[Ani Asoyan]

yes u_p is a function... sorry I changed what I wanted.

x_e is also a variable

I want to find the values of N and x which will maximize u_g function and the max value of function

and also I want to find the value of N which will maximize u_g function given the value of x (for example x=0) ..

can I do that? .. Do I have to give x_e a value ?

Answer 1

[Abdolkarim Mohammadi]

Finding the value of inputs that minimzes or maximizes the objective function value is an optimization problem. If your function is linear, then you run the following code and optimize your function:

[x, fval] = linprog (u_g, [], []); 

If your function is unimodal and relatively smooth, then you run the following code and optimize your function:

[x, fval] = fmincon (u_g, x0, [], []); 

And if the landscape of your function is unknown, i.e., you don't know whether it is linear, nonlinear, multi-modal, non-smooth, etc, then you run the following code and optimize your function:

nvars = 3;
[x, fval] = ga (u_g, nvars); 

You can refer to the documentation of each solver for more information.

[John D'Errico]

You seem to be advocating linprog for all problems. (At least those I've seen you answer.) Note that this is NOT a linear objective, so linprog is completely useless here.

[Abdolkarim Mohammadi]

I just wanted to give a general idea of the optimization tools besides the nonlinear ones that are suitale for those problems.

Answer 2

[Walter Roberson]

a=2; b=2; c=1; e=0.75 ; l=0.5;
u_g = @(x, x_e, N)(-0.5*a*x.^2+b*(x-x_e)-c*(N.^l)+e*u_p(x,x_e,N));

funmin = @(xxeN) u_g(xxeN(1), xxeN(2), xxeN(3));
funmax = @(xxeN) -u_g(xxeN(1), xxeN(2), xxeN(3));

lb = [-10 -10 -10];    %adjust as appropriate
ub = [10 10 10];       %adjust as appropriate
xxeN0 = [-.1 .2 .3];   %initial guess

[best4min, fvalmin] = fmincon(funmin, xxeN0, [], [], [], [], lb, ub);
[best4max, fvalmax] = fmincon(funmax, xxeN0, [], [], [], [], lb, ub);

Answer 3

[Cristian Garcia Milan]

I think that what you want is the function

fminbnd(fun)

that finds local minimum.

If you use

fminbnd(-fun)

you will get it max.

[Ani Asoyan]

I have to specify the range of values... how about a global minimum?

[Cristian Garcia Milan]

How about using symbolic toolbox? Then you can derivate alomg x or N and solve making equal 0

[Ani Asoyan]

you mean syms x? ,, how can I do that properly?