Minimization of Functional

| So I have a function, which is supposed to take a data matrix break it down into its respective column matrices. Then it defines a function by summing the distance from some input point, each one of these column matrices to some point X. I then nested this program so that I could attempt to minimize this function, however when I do this things start to go funny.

Firstly, often I get message,

"Warning: Gradient must be provided for trust-region
    algorithm;
    using line-search algorithm instead. 
  > In fminunc at 347
    In runMedian at 2

Local minimum possible.

fminunc stopped because the size of the current step is less than
the default value of the step size tolerance."

If I try and use fminunc instead of fminsearch. So should I be using one over the other here?

Secondly and more disturbing is that this often returns a non-sensical answer. For example, if I give it the data matrix: A=[2 -2; 0 0], and let X0=0 the answer returned to me is: X =

      0.0041   -0.0017
      0.0017    0.0042

fval =  

    0.0046

Which is completely non-sensical because since we broke A into two column matrixs, [10;0] and [-10;0] the operation of X-A(n) is not defined if X is a 2x2 matrix. The answer should be X=0 by the way. Any suggestions?

My code is:

function[X,fval] = runMedian(X0,A) %#ok<INUSD>
[X,fval] = fminsearch(@Median, X0);
  function[Y] = Median(X) 
      numberColumns = size(A,2);
      a = cell(1, numberColumns);
      for nc = 1:numberColumns
          a{nc} = A(:,nc); % Note the curly brackets for a{nc}
      end  
      for nc = 1:numberColumns 
          y(nc) = norm(X-(a{nc}));  
      end
      Y = sum(y);
  end
end

Thanks

Answer 1

[Walter Roberson]

Instead of using nested functions, please try using an anonymous function to pass the extra argument:

function[X,fval] = runMedian(X0,A)
  [X,fval] = fminsearch(@(X) Median(X,A), X0);
end

function[Y] = Median(X,A) 
  numberColumns = size(A,2);
  a = cell(1, numberColumns);
  for nc = 1:numberColumns
    a{nc} = A(:,nc); % Note the curly brackets for a{nc}
  end  
  for nc = 1:numberColumns 
    y(nc) = norm(X-(a{nc}));  
  end
  Y = sum(y);
end

I have not checked the rest of your logic.

[Walter Roberson]

There doesn't seem to be any point to going through the two independent loops or in forming the cell array. Each a{nc} is used only once, so your function can be:

numberColumns = size(A,2);

y = zeros(numberColumns,1);

for nc = 1:numberColumns

y(nc) = norm(X-A(:,nc));

end

Y = sum(y);

Or alternately, it could be the single line,

Y = sum(arrayfun(@(nc) norm(X-A(:,nc)),1:size(A,2));

[David]

Thanks for the help it got rid of the first error, however, I am guessing it may be something in minimization process that is causing the larger one because when I try and minimize your code I receive the same answer I did with my code, which should be unreasonable.

For example if I let A be [2 -2; 0 0] and try and minimize I get an answer saying X=[.0009 .0006; -.0014 .0023], which makes no sense, and if to make sure it makes no sense if you plug it back into the original function, you receive and error saying "Matrix dimensions must agree"

Thank you very much for your help.