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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Bibliographic Details
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
Online Access:Get full text
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Summary: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 (Nonlinear Anal. 55:845–858, 2003; Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets.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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-023-02269-2