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Alexander Kruger

Publications and source records attributed to Alexander Kruger.

4 recordsLinked to original sources
arXiv2026

Sequential Extremal Principle: Refinements and Applications

Sequential extremality and stationarity properties are discussed with the emphasis on those corresponding to fixed sequences of translations. Exact quantitative characterisations of the properties are provided. We show, in particular, that the sequential (as well as conventional) extremality and approximate stationarity properties possess certain stability, while the (non-approximate) stationarity does not. Dual necessary conditions for the sequential extremality and stationarity properties with fixed and non-fixed sequences of translations are established. A version of the sequential extended extremal principle is formulated. In the statements, we employ certain generalised separation conditions $(GS)$ and $(GS_\alpha)$ as well as a complementary primal-dual condition $(PD)$. To illustrate the model, we prove dual optimality/stationarity conditions for a constrained minimisation problem in which the minimal value is not necessarily attained.

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arXiv2019

Hölder Error Bounds and Hölder Calmness with Applications to Convex Semi-Infinite Optimization

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.

math.OC↗
arXiv2017

Strong Metric Subregularity of Mappings in Variational Analysis and Optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various definitions for more than two decades, it has attracted much less attention than its older "siblings", the metric regularity and the strong metric regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form f + F, where f is a function with a small calmness constant. This result is parallel to the Lyusternik-Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Frechet coderivative. Various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity.

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