How to write this small optimisation problem?

The rough sketch of the problem is:

find the maximum value of F(x,y,z)

s.t. Amin<=x<=Amax Bmin<=y<=Bmax Cmin<=z<=Cmax x,y,z belong to Real set

my approach was: I fixed z=z0. x, y are column vectors with different step sizes going from their minimum to maximum value. F is the matrix of all the different values of x,y. max(F(:)) gives me the max value. Good for two variables.

But when I have three varying variables x,y,z, I am confused how to approach it. I need to find the maximum value Fmax as well as the values of x,y,z, when F=Fmax. Trying this for a day. I do not know optimisation. Any help how to begin writing this code?

Answer 1

[Alan Weiss]

If you have Optimization Toolbox, use fmincon on -F(x,y,z) = -F(t), where t = [x,y,z]. If you just want to make a grid of points and look at the maximum on the grid:

 xgrid = linspace(Amin,Amax); % gives 100 values, change if you like
 ygrid = linspace(Bmin,Bmax);
 zgrid = linspace(Cmin,Cmax);
 % biggrid = ndgrid(xgrid,ygrid,zgrid); % for a vectorized F
 % Fvals = F(biggrid); % this works is F is vectorized. Otherwise:
 Fvals = zeros(100,100,100);
 for x = 1:100
    for y = 1:100
        for z = 1:100
            Fvals(x,y,z) = F(xgrid(x),ygrid(y),zgrid(z));
        end
    end
 end

Alan Weiss

MATLAB mathematical toolbox documentation