On the application of the SCD semismooth* newton method to variational inequalities of the second kind
- Authors: Gfrerer, Helmut , Outrata, Jiri , Valdman, Jan
- Date: 2022
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 30, no. 4 (2022), p. 1453-1484
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.
On the isolated calmness property of implicitly defined multifunctions
- Authors: Gfrerer, Helmut , Outrata, Jiri
- Date: 2023
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 30, no. 3 (2023), p. 1001-1023
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with an extension of the available theory of SCD (subspace containing derivatives) mappings to mappings between spaces of different dimensions. This extension enables us to derive workable sufficient conditions for the isolated calmness of implicitly defined multifunctions around given reference points. This stability property differs substantially from isolated calmness at a point and, possibly in conjunction with the Aubin property, offers a new useful stability concept. The application area includes a broad class of parameterized generalized equations, where the respective conditions ensure a rather strong type of Lipschitzian behavior of their solution maps. © 2023 Heldermann Verlag. All rights reserved.
Radius theorems for subregularity in infinite dimensions
- Authors: Gfrerer, Helmut , Kruger, Alexander
- Date: 2023
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 86, no. 3 (2023), p. 1117-1158
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper continues our previous work (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) on the radius of subregularity that was initiated by Asen Dontchev. We extend the results of (Dontchev et al. in Set-Valued Var Anal 28:451–473, 2020) to general Banach/Asplund spaces and to other classes of perturbations, and sharpen the coderivative tools used in the analysis of the robustness of well-posedness of mathematical problems and related regularity properties of mappings involved in the statements. We also expand the selection of classes of perturbations, for which the formula for the radius of strong subregularity is valid. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.