Strong stability in nonlinear programming revisited

The paper revisits characterizations of strong stability and strong regularity of KarushKuhn-Tucker solutions of nonlinear programs with twice differentiable data. We give a unified framework to handle both concepts simultaneousl

Calmness of constraint systems with applications

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable to detect calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk

Quantitative stability analysis of stochastic generalized equations

We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stochastic variational inequality, stationary points of classical one stage and two stage stochastic minimization problems, two stage stochastic mathematical programs with equilibrium constraints and stochastic programs with second order dominance constraints

Approximations and generalized Newton methods

We present approaches to (generalized) Newton methods in the framework of generalized equations $0\in f(x)+M(x)$, where $f$ is a function and $M$ is a multifunction. The Newton steps are defined by approximations $\hat f$ of $f$ and the solutions of $0\in \hat{f}(x)+M(x)$. We give a unified view of the local convergence analysis of such methods by connecting a certain type of approximation with the desired kind of convergence and different regularity conditions for $f+M$. Our paper is, on the one hand, thought as a survey of crucial parts of the topic, where we mainly use concepts and results of the monograph (Klatte and Kummer, Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002). On the other hand, we present original results and new features. They concern the extension of convergence results via Newton maps (Klatte and Kummer, Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002; Kummer in: Oettli, Pallaschke (eds) Advances in optimization, Springer, Berlin, 1992) from equations to generalized equations both for linear and nonlinear approximations $\hat f$, and relations between semi-smoothness, Newton maps and directional differentiability of $f$. We give a Kantorovich-type statement, valid for all sequences of Newton iterates under metric regularity, and recall and extend results on multivalued approximations for general inclusions $0\in F(x)$. Equations with continuous, non-Lipschitzian $f$ are considered, too