Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

Nguyen Ngoc Luan (), Nguyen Mau Nam (), Nguyen Nang Thieu () and Nguyen Dong Yen ()
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Nguyen Ngoc Luan: Hanoi National University of Education
Nguyen Mau Nam: Portland State University
Nguyen Nang Thieu: Vietnam Academy of Science and Technology
Nguyen Dong Yen: Vietnam Academy of Science and Technology

Journal of Optimization Theory and Applications, 2023, vol. 199, issue 2, No 12, 766-786

Abstract: 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 (Nonlinear Anal. 55:845–858, 2003; Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets.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.

Keywords: Convex polyhedron; Generalized convex polyhedron; Separation; Normal cone; Coderivative; Subdifferential; Optimal value function; 49J52; 49J53; 90C31 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02269-2

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