Using fminunc with gradient computation

Hi, I managed to find the minimizer of a function (see picture), and am now trying to include the computation of the gradient. I can't manage to do it, since the method I use to find the gradient returns an error :

Also I'm wondering why it is useful to compute the gradient in this problem?

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    % definition of x and y
    x=-10:0.1:10;
    y=-0.4:0.1:10; 

    % define a grid (x,y)
    [xx,yy]=meshgrid(x,y);

    % Evaluation of f(x,y) on this grid
    zz = f(xx,yy); 

    % Quasi Newton
    fun = @(x)(x(2) - cos(2*x(1)) - ((x(1).^2)/10)).^2 + exp((x(1).^2 + x(2).^2)/100);

    % @ = anonymous function to be used as an argument, x(1) = x, x(2) = y
    x0 = [1, 1];
    options = optimset('Display','iter', 'GradObj', 'on');

    [qN_x, qN_fval] = fminunc(fun, x0, options);

    function z = f(x,y)
    z = (y - cos(2*x) - ((x.^2)/10)).^2 + exp((x.^2 + y.^2)/100);
    end
 end

Answer 1

[Alan Weiss]

To include a gradient in the objective function, have the objective return another output, as explained in the documentation.

You seem to have two definitions of your objective function in your script. The first one ( fun = @(x)…) looks good, in that you have just one variable x. Your second, function z = f(x,y)…, is not correct in that it takes two variables. See the link above for how to write scalar objective functions.

When you are done with your objective function it should look like this;

function [z,gz] = myfun(t)
x = t(1);
y = t(2);
z = (y - cos(2*x) - ((x.^2)/10)).^2 + exp((x.^2 + y.^2)/100);
if nargout > 1 % If a gradient is called for
    gz(1) = 2*(y - cos(2*x) - ((x.^2)/10)).^2*(2*sin(2*x) - x/5)...
      + x/50*exp((x.^2 + y.^2)/100);
    gz(2) = 2*(y - cos(2*x) - ((x.^2)/10)) + y/50*exp((x.^2 + y.^2)/100);
end

I hope that I didn't ruin your homework problem by giving you too explicit an answer.

Alan Weiss

MATLAB mathematical toolbox documentation

[Torsten]

gz(1) = 2*(y - cos(2*x) - ((x.^2)/10)).*(2*sin(2*x) - x/5)...
      + x/50.*exp((x.^2 + y.^2)/100);