Metric Regularity of the Sum of Multifunctions and Applications
- Authors: Van Ngai, Huynh , Tron, Nguyen Tron , Thera, Michel
- Date: 2014
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 160, no. 2 (2014), p. 355-390
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported. © 2013 Springer Science+Business Media New York.
Metric regularity relative to a cone
- Authors: Van Ngai, Huynh , Tron, Nguyen , Théra, Michel
- Date: 2019
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 47, no. 3 (2019), p. 733-756
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The purpose of this paper is to discuss some of the highlights of the theory of metric regularity relative to a cone. For example, we establish a slope and some coderivative characterizations of this concept, as well as some stability results with respect to a Lipschitz perturbation.
Directional metric pseudo subregularity of set-valued mappings: a general model
- Authors: Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 61-87
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- Description: This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established. © 2019, Springer Nature B.V.