Let Γ := {x ∈ R n | q(x) ∈ Θ}, where q : R n → R m is a twice continuously differentiable mapping, and Θ is a nonempty polyhedral convex set in R m. In this paper, we first establish a formula for exactly computing the graphical derivative of the normal cone mapping N Γ : R n ⇒ R n , x → N Γ (x), under the condition that M q (x) := q(x) − Θ is metrically subregular at the reference point. Then, based on this formula, we exhibit formulae for computing the graphical derivative of solution mappings and present characterizations of the isolated calmness for a broad class of generalized equations. Finally, applying to optimization, we get a new result on the isolated calmness of stationary point mappings.
This paper is devoted to the study of tilt stability in finite dimensional optimization via the approach of using the subgradient graphical derivative. We establish a new characterization of tilt-stable local minimizers for a broad class of unconstrained optimization problems in terms of a uniform positive definiteness of the subgradient graphical derivative of the objective function around the point in question. By applying this result to nonlinear programming under the metric subregularity constraint qualification, we derive a second-order characterization and several new sufficient conditions for tilt stability. In particular, we show that each stationary point of a nonlinear programming problem satisfying the metric subregularity constraint qualification is a tilt-stable local minimizer if the classical strong second-order sufficient condition holds. in question. Furthermore, using this result together with a formula of Dontchev and Rockafellar [7] for the second-order limiting subdifferential of the indicator function of polyhedral convex set, they obtained a second-order characterization of tilt-stability for nonlinear programming problems with linear constraints [37, Theorem 4.5]. The main difficulty in applying the tilt-stability characterization of Poliquin and Rockafellar [37] to other nonlinear constrained optimization problems is the computation/estimation of the second-order subdifferential in terms of explicit problem data.By establishing new second-order subdifferential calculi, Mordukhovich and Rockafellar [34] derived second-order characterizations of tilt-stable minimizers for some classes of constrained optimization problems. Among other important things, they showed that for C 2 -smooth nonlinear programming problems, under the linear independence constraint qualification (LICQ), a stationary point is a tilt-stable local minimizer if and only if the strong second-order sufficient condition (SSOSC) holds. Consequently, in this setting, tiltstability is equivalent to Robinson's strong regularity [39] of the associated Karush-Kuhn-Tucker system whenever LICQ occurs at the point in question. In contrast to Robinson's strong regularity, tilt stability does not postulate LICQ as a necessary condition. This observation motivated the study of tilt-stability for nonlinear programming under constraint qualifications weaker than LICQ, aiming to cover a broader class of examined problems.Under the validity of both the Mangasarian-Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ), Mordukhovich and Outrata [31] proved that SSOSC is a sufficient condition for a stationary point to be a tilt-stable local minimizer in nonlinear programming. In [28] Mordukhovich and Nghia showed that SSOSC is indeed not a necessary condition for tilt stability and then introduced the uniform second-order sufficient condition (USOSC) to characterize tilt stability when both MFCQ and CRCQ occur. Recently, Gfrerer and Mordukhovich [12] obtained some pointbased second-order sufficient c...
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