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Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

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 (Nonlinear Anal. 55:845–858, 2003; Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets.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

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Acknowledgements

Nguyen Mau Nam would like to thank the Vietnam Institute of Mathematics-VAST (through the IM-Simons program) and the Vietnam Institute for Advanced Study in Mathematics for hospitality. Research of this author was partly supported by the USA National Science Foundation under grant DMS-2136228. Research of Nguyen Ngoc Luan and Nguyen Dong Yen was funded by the Vietnam Ministry of Education and Training under grant number B2022-CTT-06. The authors appreciate the careful review of the paper and the constructive feedback provided by the handling Associate Editor and two anonymous referees.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam

    Nguyen Ngoc Luan

  2. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, 97207, USA

    Nguyen Mau Nam

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Nguyen Nang Thieu & Nguyen Dong Yen

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  1. Nguyen Ngoc Luan
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Correspondence to Nguyen Mau Nam.

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Luan, N.N., Nam, N.M., Thieu, N.N. et al. Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets. J Optim Theory Appl 199, 766–786 (2023). https://doi.org/10.1007/s10957-023-02269-2

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