[Ipopt] Convergence issue

[Achille Sassi]

Hi, i am experiencing a strange behaviour from Ipopt:
I need to solve a relatively simple problem where i have to find the
minimum value of fuel and a control strategy which allow a vehicle to reach
a given final position.
The formulation of the problem is the following:

minimize
              J(X)

subject to
               X_L <= X <= X_U
               G_L <= G(X) <= G_U
where

- X = (L, T1, ..., T5) is the array of decision variables, which contains
the initial fuel mass L and the five control parameters T1, ..., T5.

- J(X) = X1 is the cost to be minimized and it's simply the first component
of the array X.

- G(X) is the distance between the final position of the vehicle and the
target position. G_L and G_U are not set to zero, i relaxed this constraint
by adding a small tolerance.

Loosely speaking, i am searching for the "first" feasible solution (in
terms of initial fuel load).

Now, i know that there exist a controllability threshold L* such that for
 values of L lower than L* the vehicle can't reach the target, no matter
what the control strategy is.
In fact the value of L* is exactly what i am asking Ipopt to find.
The problem is that if i set the lower bound for L simply to zero, Ipopt
doesn't converge and it just keeps iterating.
On the other hand, if i choose a lower bound which is "high enough", Ipopt
converges very quickly (10-15 iterations) to an optimal solution which
consists in choosing L equal to its lower bound.

I wrote "high enough" because i am sure that the problem lies in the
controllability threshold L*: if the lower bound for L is higher than L*
the optimal solution is obviously the closest possible value of L to L*
which leads to the saturation of its lower bound.
But if the lower bound for L is lower than L* it means that i am allowing
Ipopt to try values of L such small that the constraint on G cannot be
satisfied.

My question: isn't it strange that Ipopt can't find the optimal solution
when the decision variables can take values for which the constraint may
not be satisfied?

I hope that the description is clear enough, English is not my mother
language so i apologize for any error.
Thank you in advance for your help.
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