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Error bounds revisited

- Cuong, Nguyen, Kruger, Alexander

  • Authors: Cuong, Nguyen , Kruger, Alexander
  • Date: 2022
  • Type: Text , Journal article
  • Relation: Optimization Vol. 71, no. 4 (2022), p. 1021-1053
  • Relation: https://purl.org/au-research/grants/arc/DP160100854
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  • Description: We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

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.
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