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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 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 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.
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