We deal with a perturbed version of a Hummel-Seebeck type method to approximate a solution of variational inclusions of the form : 0 ∈ Φ(z) + F(z) where Φ is a single-valued function twice continuously Fréchet differentiable and F is a set-valued map from Rn to the closed subsets of Rn. This framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods (see [1]). We obtain, thanks to some semistability and another property (which is close to the hemistability) of the solution z ̄ of the previous inclusion, the local existence of a sequence that is superquadratically or cubically convergent.