How to minimize a function without using a loop

I have an equation with 3 constant (Nug & Sill, Lag) and 2 variable (R[ranges from 1 to 4000], alpha[ranges from 1 to 2, step of 0.1]). I need to find the value R and alpha that minimizes the function. Is there any Matlab function or any way i can do it without having to do a doublé for loop.

%q = 4000
  for i=1:q
          Theoritical(i,1) = Nug+Sill*(1-exp(-(Lag(i,1)/R)^alpha));
  end

Thank you. Darl.

Answer 1

[Guillaume]

It doesn't look like your Lag is much of a constant if there's 4000 values for it. Does Lag changes with R (since there's also 4000 R values)?

You can use fminsearch or fminbnd but if there's only 4000 values for R and 11 for alpha, you could simply calculate the result for all of them at once and get the minimum :

[R, alpha] = ndgrid(1:4000, 1:0.1:2);  %get all combinations of R and alpha
result = Nug + Sill * (1 - exp(-bsxfun(@rdivide, Lag, R) .^ alpha)); %assumes Lag is a column vector with 4000 elements
[minvalue, location] = min(result(:));
minR = R(location)
minalpha = alpha(location)

[Darlington Mensah]

Thank you @Guillaume. Lag is vector with 11 elements ranging from 0 to 4000.

[Guillaume]

Oh! I'm not sure why you accepted the answer then as it's not going to work.

What is your equation then? In your question you're indexing Lag with i, i goes from 1 to q and q is said to be 4000. That's in direct contradiction with your statement above.

How do these 11 different values figure in the equation?

[Darlington Mensah]

%q = 11
  for i=1:q
          Theoritical(i,1) = Nug+Sill*(1-exp(-(Lag(i,1)/R)^alpha));
  end

I made a mistake in my initial question. I apologize.

Per the equation, i consider each element of Lag in the computation. Omitting the q for now and assuming Lag is a constant, with variables being only alpha and R.

[Guillaume]

So, for a given R and alpha, you've 11 different theoritical (theoretical?) values. What is it you want to minimise?

[Darlington Mensah]

I have a function (ExpSemivariance) that creates a vector of 11 elements. This elements are then represented on a graph in a scattered diagram and I intend finding the curve (Theoretical) that best fit 'ExpSemivariance' by using mimimum residual

Step = 10;
q = 11;

      for i=1:q
          Theoretical(i,1) = Nug+Sill*(1-exp(-(Lag(i,1)/R)^alpha));
      end
      Error = sum(abs(Theoretical-ExpSemivariance));

      while Step > 1
          R = R + Step;
          for i=1:q
              Theoretical(i,1) = Nug+Sill*(1-exp(-(Lag(i,1)/R)^alpha));
          end
          error = sum(abs(Theoretical-ExpSemivariance));
          if error > Error
              R = R - Step;
              Step = Step/2;
          else
              Error = error;
          end
      end

The variables are R and alpha. Initially, i made alpha constant and used the code above to calculate min®. My problem is how to calculate both min® and min(alpha) for each i of the fun(Theoretical) such that Theoretical would be the best function that minimizes the error (between ExpSemivariance and Theoretical)