Directional metric regularity of multifunctions
- Authors: Ngai, Huynh Van , Thera, Michel
- Date: 2015
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 40, no. 4 (2015), p. 969-991
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.
- Description: In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity. © 2015 INFORMS.
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
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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.