Stationarity and regularity concepts for set systems
- Authors: Kruger, Alexander , University of Ballarat
- Date: 2006
- Type: Journal article
- Relation: IFIP International Federation for Information Processing
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- Description: Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle) are presented for the case of an Asplund space.
About regularity of collections of sets
- Authors: Kruger, Alexander
- Date: 2006
- Type: Text , Journal article
- Relation: Set-Valued Analysis Vol. 14, no. 2 (Jun 2006), p. 187-206
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- Description: The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular.
- Description: 2003001526
Stationarity and regularity of set systems
- Authors: Kruger, Alexander
- Date: 2005
- Type: Text , Journal article
- Relation: Pacific Journal of Optimization Vol. 1, no. 1 (2005), p. 101-126
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- Reviewed:
- Description: Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle) are presented for the case of an Asplund space.
- Description: 2003001378
Nonsmooth analysis : Weak stationarity
- Authors: Kruger, Alexander
- Date: 2009
- Type: Text , Book chapter
- Relation: Encyclopedia of Optimization Chapter p. 2658-2664
- Full Text: false
- Description: The article considers stationary and regularity concepts for extended real-valued functions on metric spaces.
Stationarity and regularity of infinite collections of sets
- Authors: Kruger, Alexander , López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 154, no. 2 (2012), p. 339-369
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC.
About stationarity and regularity in variational analysis
- Authors: Kruger, Alexander
- Date: 2009
- Type: Text , Journal article
- Relation: Taiwanese Journal of Mathematics Vol. 13, no. 6A (2009), p. 1737-1785
- Full Text: false
- Description: Stationarity and regularity concepts for the three typical for variational analysis classes of objects - real-valued functions, collections of sets, and multifunctions - are investigated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity properties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed.
Stationarity and regularity of real-valued functions
- Authors: Kruger, Alexander
- Date: 2006
- Type: Text , Journal article
- Relation: Applied and Computational Mathematics Vol. 5, no. 1 (2006), p. 79-93
- Full Text: false
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- Description: Different stationarity and regularity concepts for extended real-valued functions on metric spaces are considered in the paper. The properties are characterized in terms of certain local constants. A classifcation scheme for stationarity/regularity constants and corresponding concepts is proposed. The relations between different constants are established.
- Description: C1
- Description: 2003001544
Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization
- Authors: Kruger, Alexander , López, Marco
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 155, no. 2 (2012), p. 390-416
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements-normals and/or subdifferentials.