A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

Abstract

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.

Combining stabilized SQP with the augmented Lagrangian algorithm

“…in contradiction with (26). Thus, the sequence {ξ j | j ∈ J} cannot have the zero accumulation point.…”

“…In fact, this error bound is equivalent to the assumption that the multiplier (λ,μ) ∈ M(x) is noncritical, as defined in [30]; see also [26] and [31,Section 1.3]. As in what follows we would only invoke this notion for the equality-constrained case, we give here the corresponding simpler definition.…”

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

“…In the terminology of [21, Definition 1.41] (see also [19,15]), a multiplier λ * associated to a primal solution x * of the KKT system (6.1), is called noncritical if there exists no pair (ξ, η) ∈ R n × R m , with ξ = 0, satisfying (6.4)…”

“…It can be easily shown that if SOC (6.3) holds, then λ * is automatically noncritical; see [19,15,21] for details. Also, it should be emphasized that usually, the class of noncritical multipliers is much wider than the class of multipliers satisfying SOC.…”

“…In particular, we show that the methods under consideration converge quadratically to a solution of (1.1) under the only assumption that the complementarity system satisfies a mild piecewise local error bound condition. In section 6, the special case of KKT systems arising from optimization or variational problems is considered, and it is shown that the noncriticality assumption on the Lagrange multiplier [19,15] is sufficient for convergence in this case. Sections 7 and 8 consider different reformulations of KKT systems associated to generalized Nash equilibrium problems, where we exhibit sufficient conditions for the required error bounds in that setting.…”