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Quasi-relative interiors for graphs of convex set-valued mappings

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Abstract

This paper aims at providing further studies of the notion of quasi-relative interior for convex sets. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.

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Acknowledgements

The authors are indebted to Nicolas Hadjisavvas, Pedro Pérez-Aros, Constantin Zălinescu, and two anonymous referees for their careful reading of the original version of the paper with helpful remarks and suggestions that allowed us to significantly improve the obtained results.

Funding

B. S. Mordukhovich: Research of this author was partly supported by the USA National Science Foundation under Grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research Grant #15RT04, and by the Australian Research Council under Discovery Project DP-190100555. N. M. Nam: Research of this author was partly supported by the USA National Science Foundation under Grant DMS-1716057.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Natural Sciences, Duy Tan University, Da Nang, Vietnam

    Dang Van Cuong

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich

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

    Nguyen Mau Nam

Authors
  1. Dang Van Cuong
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  2. Boris S. Mordukhovich
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  3. Nguyen Mau Nam
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Correspondence to Nguyen Mau Nam.

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Dedicated to Nicolas Hadjisavvas on the occasion of his 65th birthday.

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Van Cuong, D., Mordukhovich, B.S. & Nam, N.M. Quasi-relative interiors for graphs of convex set-valued mappings. Optim Lett 15, 933–952 (2021). https://doi.org/10.1007/s11590-019-01447-4

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  • DOI: https://doi.org/10.1007/s11590-019-01447-4

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