The approach I finally followed is the following:
I used NLopt for the optimization and the objective function was constructed to compute the squared error of the problem.
The algorithm that showed the most promising results was COBYLA (Local derivative-free optimization). It supports box constraints and non-linear constraints. The linear inequity constraints were introduced as non-linear constraints, which should be generally feasible.
Simple benchmarking shows that it does converge a little slower than a Lev-Mar approach, but speed is sacrificed due to the need for constraints.
