Active constraints in fgoalattain result, but I didn't provide constraints.

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?

Answer 1

[Alan Weiss]

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

[Kevin Doherty]

Ah, this makes sense to me now.

Also worth noting is that there are four active inequalities because I set "GoalsExactAchieve" to 2. This is due (I think) to the fact that once a goal is achieved, this counts as an active inequality, and as long as it is not overachieved (within TolCon), this counts as another active inequality. For example, if I set "GoalsExactAchieve" to 0 then there are two active inequalities.

Thanks Alan