MATLAB: Active constraints in fgoalattain result, but I didn’t provide constraints.

constraintsfgoalattainoptimisation

I am using the optimisation toolbox in R2008b. I have noticed that when I run fgoalattain with no constraints, some constraints appear to be violated throughout iterations of the algorithm and the algorithm returns a solution with active nonlinear inequalities. My understanding is that this should not happen, since I have not provided any constraints, but perhaps there is a good reason for it.
Here is an example:
function x = test
options = optimset('Display','iter',...
    'GoalsExactAchieve',2);
weight = [1,1];
goal = [2,2];
x0 = [1,2];
x = fgoalattain(@FitF,x0,goal,weight,[],[],[],[],[],[],[],options);
function ObjF = FitF(x)
ObjF = [exp(2*x(2))-exp(x(1));
    exp(x(1))-2*exp(x(2))];
In the above code, I am optimising a relatively simple set of equations and am providing no linear or nonlinear constraints. This returns the following output:
>> x = test
                   Attainment        Max     Line search     Directional 
   Iter F-count        factor    constraint   steplength      derivative   Procedure 
      0      4              0       49.8799                                            
      1      8          2.114         8.168            1           0.704     
      2     12          2.451         1.567            1           0.615    Hessian modified twice  
      3     16     1.906e-008        0.8339            1          -0.984     
      4     20              0        0.0666            1       -1.3e-007     
      5     24     1.388e-017     0.0005193            1       1.04e-015     
      6     28              0    3.063e-008            1      -1.38e-013    Hessian modified  
  Optimization terminated: magnitude of search direction less than 2*options.TolX
   and maximum constraint violation is less than options.TolCon.
  Active inequalities (to within options.TolCon = 1e-006):
    lower      upper     ineqlin   ineqnonlin
                                       1
                                       2
                                       3
                                       4
x =
      2.1368    1.1744
>>
As you can see, fgoalattain works just fine. But why is Max constraint non-zero and why does my solution have active nonlinear inequalities when I have not provided any nonlinear constraints?

Best Answer

I think that what is happening is that fgoalattain internally reformulates your problem as described in the documentation, and converts the problem to a standard minimization problem with nonlinear constraints:
min γ
x,γ
such that
 F(x) – w·γ ≤ F*.
I believe that the exit message is referring to something like these nonlinear constraints, which are not quite as simple as I just described, but you can read more details in the documentation link I gave.
Alan Weiss
MATLAB mathematical toolbox documentation
Related Question