MATLAB: Problem with optimization using fminunc + GradObj

# gradobjfminuncMATLABoptimization

Hello,
I have problem with optimization using fminunc with GradObj set to on. I'm trying to find minimum in White & Holst function – which have only one minimum at [1.0, 1.0]. Without GradObj fminunc is able to find optimum. With GradObj function stops at [0.8043, 0.5152] with message:
—–
                                                                                                                                        First-order 
 Iteration  Func-count       f(x)        Step-size       optimality
     0           1          749.038                     2.36e+003
     1           2           61.637     0.00042391            157  
     2           4          1.60037             10           13.5  
     3           5           1.1263              1              2  
     4           6          1.12404              1              2  
     5           7          1.11603              1           1.99  
     6           8          1.09697              1            2.5  
     7           9          1.04559              1           4.69  
     8          10         0.949647              1           8.73  
     9          11         0.878371              1           10.3  
    10          12         0.637169              1           8.25  
    11          13         0.395188              1           2.76  
    12          15         0.289895       0.414369           2.66  
    13          17         0.288287       0.190601           4.45  
    14          18         0.260038              1           5.08  
    15          19            0.172              1           3.41  
    16          20         0.107135              1          0.765  
    17          22         0.100943       0.260267           2.38  
    18          23        0.0843457              1           3.47  
    19          24        0.0516633              1            0.9  
                                                        First-order 
 Iteration  Func-count       f(x)        Step-size       optimality
    20          25        0.0460001              1           1.11  
    21          26        0.0431929              1          0.123  
    22          27        0.0413137              1        0.00751  
    23          28        0.0409519              1        0.00349  
    24          29        0.0409373              1       7.4e-006
Local minimum found.
Optimization completed because the size of the gradient is less than the default value of the function tolerance.
<stopping criteria details>
 X =
    0.8043    0.5152
 FVAL =
    0.0409
 EXITFLAG =
     1
 OUTPUT = 
       iterations: 24
        funcCount: 29
         stepsize: 1
    firstorderopt: 7.4034e-006
        algorithm: 'medium-scale: Quasi-Newton line search'
          message: [1x438 char]
—–
Here is code I'm running:
function [f,G] = func2(x)
    f = 100*(x(2) - x(1)*x(1)*x(1))*(x(2) - x(1)*x(1)*x(1)) + (1 - x(1))*(1 - x(1));
    % Gradient of the objective function
    if nargout > 1
        G = [-600*x(2)*x(1)*x(1) + 600*x(1)*x(1)*x(1)*x(1)*x(1) - 2
                200*x(2) - 200*x(1)*x(1)*x(1) + 2*x(2)];
    end
end
…
options = optimset('LargeScale', 'off', 'InitialHessType', 'Scaled-Identity', 'GradObj', 'on', 'Display', 'iter');
% where startX = -1.2 and startY = 1.0
[X,FVAL,EXITFLAG,OUTPUT] = fminunc(@func2,[startX, startY], options)
Please, help me solve this problem.

Best Answer

You have more errors in your gradient. The correct gradient is
G = [-600*x(2)*x(1)^2 + 600*x(1)^5 - 2+2*x(1)
              200*x(2) - 200*x(1)^3];
Note, by the way, how much the use of the power symbol ^ reduces the length of the code.
(Edited to correct errors of my own).
Note that you can check gradients using symbolic algebra, e.g.:
G = 100*(y - x^3)^2 + (1 - x)^2;
dGx = expand(diff(G,x))
dGy = expand(diff(G,y))
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