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We propose and analyze a perturbed version of the Josephy-Newton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, quasi-Newton sequential quadratic programming, sequential quadratically constrained quadratic programming, linearly constrained Lagrangian methods, etc. For the linearly constrained Lagrangian methods, we obtain superlinear convergence under the assumptions weaker than previously used in the literature. Another possible application of the general framework is concerned with the development of truncated versions of sequential quadratic programming.