We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush-Kuhn-Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.

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We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving sharp local convergence results for some constrained optimization algorithms under the reduced smoothness hypotheses. Specifically, we consider applications to the augmented Lagrangian method and to the linearly constrained Lagrangian method for problems with Lipschitzian derivatives but possibly without second derivatives, and our local convergence analysis for these methods improves all the existing theories of this kind.