Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a polyhedral convex set obtained by Ng et al. (Nonlinear...

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Published in:	Journal of optimization theory and applications Vol. 199; no. 2; pp. 766 - 786
Main Authors:	Luan, Nguyen Ngoc, Nam, Nguyen Mau, Thieu, Nguyen Nang, Yen, Nguyen Dong
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.11.2023 Springer Nature B.V
Subjects:	Applications of Mathematics Calculus Calculus of Variations and Optimal Control; Optimization Cones Convex analysis Convexity Engineering Mathematical functions Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Polyhedra Regular Paper Theory of Computation 90C31 Optimal value function Separation 49J53 Normal cone 49J52 Convex polyhedron Generalized convex polyhedron Coderivative Subdifferential
ISSN:	0022-3239, 1573-2878
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Abstract	In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a polyhedral convex set obtained by Ng et al. (Nonlinear Anal. 55:845–858, 2003; Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets. We also introduce and study the notions of generalized polyhedral multifunctions and optimal value functions generated by generalized polyhedral convex multifunctions along with their generalized differentiation calculus rules.
AbstractList	In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a polyhedral convex set obtained by Ng et al. (Nonlinear Anal. 55:845–858, 2003; Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets. We also introduce and study the notions of generalized polyhedral multifunctions and optimal value functions generated by generalized polyhedral convex multifunctions along with their generalized differentiation calculus rules. In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a polyhedral convex set obtained by Ng et al. (Nonlinear Anal. 55:845–858, 2003; Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets. We also introduce and study the notions of generalized polyhedral multifunctions and optimal value functions generated by generalized polyhedral convex multifunctions along with their generalized differentiation calculus rules.
Author	Luan, Nguyen Ngoc Thieu, Nguyen Nang Yen, Nguyen Dong Nam, Nguyen Mau
Author_xml	– sequence: 1 givenname: Nguyen Ngoc surname: Luan fullname: Luan, Nguyen Ngoc organization: Department of Mathematics and Informatics, Hanoi National University of Education – sequence: 2 givenname: Nguyen Mau orcidid: 0000-0003-4850-2301 surname: Nam fullname: Nam, Nguyen Mau email: mnn3@pdx.edu organization: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University – sequence: 3 givenname: Nguyen Nang surname: Thieu fullname: Thieu, Nguyen Nang organization: Institute of Mathematics, Vietnam Academy of Science and Technology – sequence: 4 givenname: Nguyen Dong surname: Yen fullname: Yen, Nguyen Dong organization: Institute of Mathematics, Vietnam Academy of Science and Technology
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Cites_doi	10.1007/978-3-642-02431-3 10.1007/BF01581072 10.1016/j.na.2003.07.008 10.1007/BFb0120929 10.1007/978-3-642-54265-7 10.1080/02331934.2015.1032284 10.1007/978-3-662-02796-7 10.1080/01630563.2017.1387863 10.1007/s11228-009-0120-5 10.1007/978-3-031-26458-0 10.1515/9781400873173 10.1007/978-1-4612-1394-9 10.1007/s11228-022-00647-y 10.1137/060658576 10.1023/A:1022988116044 10.1080/02331934.2019.1614179 10.1007/978-3-030-94785-9
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Snippet	In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample...
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SubjectTerms	Applications of Mathematics Calculus Calculus of Variations and Optimal Control; Optimization Cones Convex analysis Convexity Engineering Mathematical functions Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Polyhedra Regular Paper Theory of Computation
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Title	Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets
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