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Helmut Gfrerer

Publications and source records attributed to Helmut Gfrerer.

35 records · Page 2Linked to original sources
arXiv2019

Stability analysis for parameterized variational systems with implicit constraints

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non-restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples

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arXiv2019

On estimating the regular normal cone to constraint systems and stationarity conditions

Estimating the regular normal cone to constraint systems plays an important role for the derivation of sharp necessary optimality conditions. We present two novel approaches and introduce a new stationarity concept which is stronger than M-stationarity. We apply our theory to three classes of mathematical programs frequently arising in the literature.

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arXiv2018

On the Aubin property of solution maps to parameterized variational systems with implicit constraints

In the paper a new sufficient condition for the Aubin property to a class of parameterized variational systems is derived. In these systems the constraints depend both on the parameter as well as on the decision variable itself and they include, e.g., parameter-dependent quasi-variational inequalities and implicit complementarity problems. The result is based on a general condition ensuring the Aubin property of implicitly defined multifunctions which employs the recently introduced notion of the directional limiting coderivative. Our final condition can be verified, however, without an explicit computation of these coderivatives. The procedure is illustrated by an example.

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arXiv2018

Characterizations of Tilt-Stable Minimizers in Second-Order Cone Programming

This paper is devoted to the study of tilt stability of local minimizers, which plays an important role in both theoretical and numerical aspects of optimization. This notion has been comprehensively investigated in the unconstrained framework as well as for problems of nonlinear programming with $C^2$-smooth data. Available results for nonpolyhedral conic programs were obtained only under strong constraint nondegeneracy assumptions. Here we develop an approach of second-order variational analysis, which allows us to establish complete neighborhood and pointbased characterizations of tilt stability for problems of second-order cone programming generated by the nonpolyhedral second-order/Lorentz/ice-cream cone. These characterizations are established under the weakest metric subregularity constraint qualification condition.

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arXiv2018

Linearized M-stationarity conditions for general optimization problems

This paper investigates new first-order optimality conditions for general optimization problems. These optimality conditions are stronger than the commonly used M-stationarity conditions and are in particular useful when the latter cannot be applied because the underlying limiting normal cone cannot be computed effectively. We apply our optimality conditions to a MPEC to demonstrate their practicability.

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arXiv2017

Calculus for directional limiting normal cones and subdifferentials

The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

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arXiv2017

Second-Order Variational Analysis of Parametric Constraint and Variational Systems

This paper is devoted to the generalized differential study of the normal cone mappings associated with a large class of parametric constraint systems (PCS) that appear, in particular, in nonpolyhedral conic programming. Conducting a local second-order analysis of such systems, we focus on computing the (primal-dual) graphical derivative of the normal cone mapping under the $C^2$-cone reducibility of the constraint set together with the fairly weak metric subregularity constraint qualification and its uniform parametric counterpart known as Robinson stability. The obtained precise formulas for computing the underlying second-order object are applied to the derivation of comprehensive conditions ensuring the important stability property of isolated calmness for solution maps to parametric variational systems associated with the given PCS.

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arXiv2017

On the Aubin property of a class of parameterized variational systems

The paper deals with a new sharp criterion ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class includes parameter-dependent variational inequalities with non-polyhedral constraint sets and also parameterized generalized equations with conic constraints. The new criterion requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested procedure is illustrated by means of examples.

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arXiv2016

Robinson Stability of Parametric Constraint Systems via Variational Analysis

This paper investigates a well-posedness property of parametric constraint systems named here Robinson stability. Based on advanced tools of variational analysis and generalized differentiation, we derive first-order and second-order conditions for this property under minimal constraint qualifications and establish relationships of Robinson stability with other well-posedness properties in variational analysis and optimization. The results obtained are applied to robust Lipschitzian stability of parametric variational systems.

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arXiv2016

An SQP method for mathematical programs with vanishing constraints with strong convergence properties

We propose an SQP algorithm for mathematical programs with vanishing constraints which solves at each iteration a quadratic program with linear vanishing constraints. The algorithm is based on the newly developed concept of $\mathcal Q$-stationarity [5]. We demonstrate how $\mathcal Q_M$-stationary solutions of the quadratic program can be obtained. We show that all limit points of the sequence of iterates generated by the basic SQP method are at least M-stationary and by some extension of the method we also guarantee the stronger property of $\mathcal Q_M$-stationarity of the limit points.

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arXiv2016

New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints

In this paper we consider a sufficiently broad class of nonlinear mathematical programs with disjunctive constraints, which, e.g., include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of ${\mathcal Q}$-stationarity as introduced in the recent paper [2]. ${\mathcal Q}$-stationarity can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called ${\mathcal Q}_M$-stationarity. We show how the property of ${\mathcal Q}_M$-stationarity (and thus also of M-stationarity) can be efficiently verified for the considered problem class by computing ${\mathcal Q}$-stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for ${\mathcal Q}_M$-stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.

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arXiv2016

Lipschitz and Hölder stability of optimization problems and generalized equations

This paper studies stability aspects of solutions of parametric mathematical programs and generalized equations, respectively, with disjunctive constraints. We present sufficient conditions that, under some constraint qualifications ensuring metric subregularity of the constraint mapping, continuity results of upper Lipschitz and upper Hölder type, respectively, hold. Furthermore, we apply the above results to parametric mathematical programs with equilibrium constraints and demonstrate, how some classical results for the nonlinear programming problem can be recovered and even improved by our theory.

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arXiv2016

Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints

We consider optimization problems with a disjunctive structure of the constraints. Prominent examples of such problems are mathematical programs with equilibrium constraints or vanishing constraints. Based on the concepts of directional subregularity and their characterization by means of objects from generalized differentiation, we obtain the new stationarity concept of extended M-stationarity, which turns out to be an equivalent dual characterization of B-stationarity. These results are valid under a very weak constraint qualification of Guignard type which is usually very difficult to verify. We also state a new constraint qualification which is a little bit stronger but verifiable. Further we present second-order optimality conditions, both necessary and sufficient. Finally we apply these results to the special case of mathematical programs with equilibrium constraints and compute explicitly all the objects from generalized differentiation. For this type of problems we also introduce the concept of strong M-stationarity which builds a bridge between S-stationarity and M-stationarity.

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arXiv2016

On Lipschitzian properties of implicit multifunctions

The paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool one employs the directional limiting coderivative which, together with the graphical derivative, enable us a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreover, as an auxiliary tool, a handy formula for the computation of the directional limiting coderivative of the normal-cone map with a polyhedral set has been derived which perfectly matches the framework of [11]. All important statements are illustrated by examples.

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arXiv2016

Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions

This paper is devoted to the study of tilt stability of local minimizers for classical nonlinear programs with equality and inequality constraints in finite dimensions described by twice continuously differentiable functions. The importance of tilt stability has been well recognized from both theoretical and numerical perspectives of optimization, and this area of research has drawn much attention in the literature, especially in recent years. Based on advanced techniques of variational analysis and generalized differentiation, we derive here complete pointbased second-order characterizations of tilt-stable minimizers entirely in terms of the initial program data under the new qualification conditions, which are the weakest ones for the study of tilt stability.

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arXiv2016

New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis

In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. Compared with the usual way of formulating MPEC through a KKT condition, this formulation has the advantage that it does not involve extra multipliers as new variables, and it usually requires weaker assumptions on the problem data. Using the so-called first order sufficient condition for metric subregularity, we derive verifiable sufficient conditions for the metric subregularity of the involved set-valued mapping, or equivalently the calmness of the perturbed generalized equation mapping.

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