# Minimize function with respect to multiple variables

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Nick Klodowski on 10 Nov 2011
Commented: Walter Roberson on 3 Apr 2019
Hi,
I have a function f(b1,b2,b3,x,y1,y2,y3) that requires multiple inputs. How can I find the values of b1, b2, and b3 that minimize this function for given values of x, y1, y2, and y3?
Thanks for the help.

Jonathan on 10 Nov 2011
You can use the function fminsearch for this, which requires an initial guess. Here is how it might look for you.
x = 1;
y1 = 2;
y2 = 3;
y3 = 4;
fun = @(b) f(b(1), b(2), b(3), x, y1, y2, y3);
b_guess = [10 20 30];
b_min = fminsearch(fun, b_guess);
b1_min = b_min(1);
b2_min = b_min(2);
b2_min = b_min(3);
~Jonathan
Walter Roberson on 3 Apr 2019
fminsearch() only accepts a single vector as input. You need to construct a 1 x (3*n) vector as your input. Your code can reshape and extract portions as needed.

Mohamad Alsioufi on 9 Dec 2017
Edited: Mohamad Alsioufi on 9 Dec 2017
Hi there, I have the same problem, but when I tried to use 'fminsearch' function I had some problems, the following code is a part of my function myGP(x,y,x_star,y_hat):
segmaSE = 1;
lengthSE = 1;
theta0 =[segmaSE lengthSE];
kernelFunc = @(x1,x2,theta)(theta(1)^2)*exp(-0.5*(pdist2(x1/theta(2), x2/theta(2)).^2));
marginal_likelihood =@(y,x,theta,N) -0.5*y'*pinv(kernelFunc(x,x,theta))*y - 0.5*log(abs(kernelFunc(x,x,theta))) - N*.5 * log (2*pi);
fun = @(theta)marginal_likelihood(y,x,theta,N);
theta0 =[segmaSE lengthSE];
theta=fminsearch(fun,theta0);
However I got the following error:
Assignment has more non-singleton rhs dimensions than non-singleton subscripts
Error in fminsearch (line 200) fv(:,1) = funfcn(x,varargin{:});
Error in myGP (line 21) theta=fminsearch(fun,theta0);
Walter Roberson on 9 Dec 2017
You do not give us any information about the sizes of the variables, which makes it difficult to test.
I notice that you always call kernelFunc() with (x, x, theta). If x is scalar or row vector then the result of the pdist2() call will be 0. If x is N x M for N > 1 then the result of the pdist2() will be N x N. The exp() will not change that, and multiplying by the scalar will not change that.
pinv() of N x N will be N x N . In order for pinv()*y to work, y must be N x P for some P, with the * giving an N x P result. The y' * before that would be * of a P x N, so that would be P x N * N * P, giving a P x P result. You multiply that by -0.5 and you subtract 0.5*log(abs(kernelFunc(x,x,theta))) where we have already determined that the kernelFunc returns an N x N result. P x P minus N x N will work under the circumstance that P is 1 or P is N; either way the subtraction going to return a N x N result .
Now, the objective function return value for fminsearch needs to be a scalar. For N x N to be scalar, then x would have had to have been N x M = 1 x M -- but in that case the pdist2() would return 0 making it essentially useless to make the pdist2() call.
This suggests that your formula is either fundamentally incorrect (but it does not look to be wrong), or else that you are asking for a matrix minimization, which fminsearch() cannot handle.