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

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Optimization Vol. 69, no. 7-8 (2020), p. 1681-1701
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: 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. © 2019, © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
  • Description: The research of the first author was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the second author was supported by the Grant Agency of the Czech Republic, Project 17-04301S and the Australian Research Council, Project 10.13039/501100000923DP160100854.

On lipschitzian properties of implicit multifunctions

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2016
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 26, no. 4 (2016), p. 2160-2189
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: This paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool we employ the directional limiting coderivative which, together with the graphical derivative, enables 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 [A. L. Dontchev and R. T. Rockafellar, SIAM J. Optim., 6 (1996), pp. 1087{1105]. All important statements are illustrated by examples. © 2016 Society for Industrial and Applied Mathematics.

On the Aubin property of a class of parameterized variational systems

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2017
  • Type: Text , Journal article
  • Relation: Mathematical Methods of Operations Research Vol. 86, no. 3 (2017), p. 443-467
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper deals with a new sharp condition ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class encompasses various types of parameterized variational inequalities/generalized equations with fairly general constraint sets. The new condition 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. © 2017, Springer-Verlag GmbH Germany.

On computation of limiting coderivatives of the normal-cone mapping to inequality systems and their applications

- Gfrerer, Helmut, Outrata, Jiri

Calculus for directional limiting normal cones and subdifferentials

- Benko, Matúš, Gfrerer, Helmut, Outrata, Jiri

  • Authors: Benko, Matúš , Gfrerer, Helmut , Outrata, Jiri
  • Date: 2019
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 27, no. 3 (2019), p. 713-745
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  • Description: 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.

On computation of generalized derivatives of the normal-cone mapping and their applications

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2016
  • Type: Text , Journal article
  • Relation: Mathematics of Operations Research Vol. 41, no. 4 (2016), p. 1535-1556
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  • Description: The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the normal-cone mapping related to C2 inequality constraints under very weak qualification conditions. This enables us to provide the graphical derivative and the regular coderivative of the solution map to a class of parameterized generalized equations with the constraint set of the investigated type. On the basis of these results, we finally obtain a characterization of the isolated calmness property of the mentioned solution map and derive strong stationarity conditions for an MPEC with control constraints. © 2016 INFORMS.

On a semismooth* Newton method for solving generalized equations

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2021
  • Type: Text , Journal article
  • Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 489-517
  • Relation: https://doi.org/10.1137/19M1257408
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  • Description: 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 multivalued part of the considered GE. The method is based on the new notion of semismoothness\ast, which, together with a suitable regularity condition, ensures 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. © 2021 Society for Industrial and Applied Mathematics

Stability analysis for parameterized variational systems with implicit constraints

- Benko, Matus, Gfrerer, Helmut, Outrata, Jiri

  • Authors: Benko, Matus , Gfrerer, Helmut , Outrata, Jiri
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 167-193
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  • Description: 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. © 2019, The Author(s).

The radius of metric subregularity

- Dontchev, Asen, Gfrerer, Helmut, Kruger, Alexander, Outrata, Jiri

  • Authors: Dontchev, Asen , Gfrerer, Helmut , Kruger, Alexander , Outrata, Jiri
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 3 (2020), p. 451-473, http://purl.org/au-research/grants/arc/DP160100854
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  • Description: There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings. © 2019, Springer Nature B.V.
  • Description: Funding details: Austrian Science Fund, FWF, P26132-N25, P26640-N25, P29190-N32 Funding details: National Science Foundation, NSF Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP160100854 Funding details: Austrian Science Fund, FWF Funding details: Universiteit Stellenbosch, US, P26640-N25 P26132-N25, BodyRef/PDF/11228_2019_Article_523.pdf Funding details: Grantová Agentura

On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives

- Gfrerer, Helmut, Outrata, Jiri

  • Authors: Gfrerer, Helmut , Outrata, Jiri
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Journal of Mathematical Analysis and Applications Vol. 508, no. 2 (2022), p.
  • Relation: http://purl.org/au-research/grants/arc/DP160100854
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  • Description: The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the graphically Lipschitzian mappings and thus a number of multifunctions, frequently arising in optimization and equilibrium problems. The developed theory makes use of new generalized derivatives, provides us with some calculus rules and reveals a number of interesting connections. In particular, it enables us to construct a modification of the semismooth* Newton method with improved convergence properties and to derive a generalization of Clarke's Inverse Function Theorem to multifunctions together with new efficient characterizations of strong metric (sub)regularity and tilt stability. © 2021 The Author(s)
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