arXiv subjects

H. Gfrerer

Publications and source records attributed to H. Gfrerer.

7 recordsLinked to original sources

arXiv2026

Second-order optimality conditions for optimization problems with generalized equation constraints

This paper provides second-order optimality conditions for optimization problems with generalized equation constraints (GEPs), a framework that encompasses several important and challenging models in mathematical programming, including mathematical programs with variational inequality constraints (MPVIs) and bilevel programs. The obtained optimality conditions are novel even for these particular problem classes. As an application, second-order optimality conditions for MPVIs are detailed. The technical key lies in developing first- and second-order variational analysis of the highly intricate constraint system, which is needed to capture the local curvature of the feasible set entering these optimality conditions. Part of this task was already carried out in our companion paper \cite{BeGfrYeZhangZhou}, and here we complete the study. Comprehensive variational analysis results are derived, which are of independent interest.

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arXiv2026

On the application of the SCD semismooth* Newton method to solving Stokes problem with stick-slip boundary conditions

The paper deals with the 3D Stokes problem with Navier-Tresca stick-slip boundary conditions. A weak formulation of this problem leads to a variational inequality of the second kind, coupled with an equality constraint. This problem is then approximated using the mixed finite element method, yielding a generalized equation, to the numerical solution of which we implement a variant of the SCD semismooth* Newton method. This includes also a globalization technique ensuring convergence for arbitrary starting points. Numerical experiments demonstrate the effeciency of this approach.

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arXiv2024

On the role of semismoothness in nonsmooth numerical analysis: Theory

For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth$^*$ multifunctions.

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arXiv2024

On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimization

In this paper we present GSSN, a globalized SCD semismooth* Newton method for solving nonsmooth nonconvex optimization problems. The global convergence properties of the method are ensured by the proximal gradient method, whereas locally superlinear convergence is established via the SCD semismooth* Newton method under quite weak assumptions. The Newton direction is based on the SC (subspace containing) derivative of the subdifferential mapping and can be computed by the (approximate) solution of an equality-constrained quadratic program. Special attention is given to the efficient numerical implementation of the overall method.

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arXiv2023

Hoffman constant of the argmin mapping in linear optimization

The main goal of this paper is to provide a point-based expression for the Hoffman constant of the argmin mapping in linear optimization, understood as the sharp Lipschitz constant restricted to its domain. The work is mainly developed in the parametric context of right-hand side perturbations of the constraint system. To the authors' knowledge, this is the first exact formula for this constant, although we can find in the literature different upper estimates. The paper tackles this objective from a broader perspective, which introduces new tools of their own interest, such as the concept of well-connected piecewise convex mapping. We isolate the nice behavior of such mappings to derive a crucial equality between the Hoffman constant (which is a global stability measure) and the supremum of calmness moduli (of local nature). The paper also includes some specifics about directional stability of optimal solutions and finishes with some conclusions and notes about further research.

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arXiv2022

On the SCD semismooth* Newton method for generalized equations with application to a class of static contact problems with Coulomb friction

In the paper, a variant of the \ssstar Newton method is developed for the numerical solution of generalized equations, in which the multi-valued part is a so-called SCD (subspace containing derivative) mapping. Under a rather mild regularity requirement, the method exhibits (locally) superlinear convergence behavior. From the main conceptual algorithm, two implementable variants are derived whose efficiency is tested via a generalized equation modeling a discretized static contact problem with Coulomb friction.

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arXiv2019

On a semismooth* Newton method for solving generalized equations

In the paper, a Newton-type method for the solution of generalized equations (GEs) is derived, where the linearization concerns both the single-valued and the multi-valued part of the considered GE. The method is based on the new notion of semismoothness${}^*$ which, together with a suitable regularity condition, ensure the local superlinear convergence. An implementable version of the new method is derived for a class of GEs, frequently arising in optimization and equilibrium models.

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