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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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References

  1. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  Google Scholar 

  3. Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont, MA (2003)

    MATH  Google Scholar 

  4. Borwein, J.M., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)

    Article  MathSciNet  Google Scholar 

  5. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, Part I: quasi-relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  Google Scholar 

  6. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd edn. Springer, New York (2006)

    Book  Google Scholar 

  7. Borwein, J.M., Lucet, Y., Mordukhovich, B.S.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 7, 375–393 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Boţ, R.I., Csecnet, E.R., Wanka, G.: Regularity condition via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)

    Article  MathSciNet  Google Scholar 

  9. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)

    Book  Google Scholar 

  10. Daniele, P., Giuffré, S., Idone, G., Maugeri, A.: Infinite dimensional duality and applications. Math. Ann. 339, 221–239 (2007)

    Article  MathSciNet  Google Scholar 

  11. Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers and \(\varepsilon \)-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. J. Optim. Theory Appl. 145, 196–211 (2010)

    Article  MathSciNet  Google Scholar 

  12. Flores-Bazan, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)

    Article  MathSciNet  Google Scholar 

  13. Flores-Bazan, Fabian, Flores-Bazan, Fernando, Laengle, S.: Characterizing effciency on infinite-dimensional commodity spaces with ordering cones having possibly empty interior. J. Optim. Theory Appl 164, 455–478 (2015)

    Article  MathSciNet  Google Scholar 

  14. Flores-Bazan, F., Mastroeni, G., Vera, C.: Proper or weak efficiency via saddle point conditions in cone-constrained nonconvex vector optimization. J. Optim. Theory Appl. 181, 787–816 (2019)

    Article  MathSciNet  Google Scholar 

  15. Giannessi, F.: Constrained Optimization and Image Space Analysis, I: Separation of Sets and Optimality Conditions. Springer, Berlin (2005)

    MATH  Google Scholar 

  16. Hadjisavvas, N., Pardalos, P.M.: Advances in Convex Analysis and Global Optimization; Honoring the Memory of C. Caratheodory (1873–1950). Kluwer Academic Publishers, Dordrecht (2001)

    Book  Google Scholar 

  17. Hadjisavvas, N., Schaible, S.: Quasimonotone variational inequalities in Banach spaces. JOTA 90, 95–111 (1996)

    Article  MathSciNet  Google Scholar 

  18. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, I: Fundamentals. Springer, Berlin (1993)

    Book  Google Scholar 

  19. Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York (1975)

    Book  Google Scholar 

  20. Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)

    MATH  Google Scholar 

  21. Kusraev, A.G., Kutateladze, S.S.: Subdifferentials: Theory and Applications. Kluwer, Dordrecht (1995)

    Book  Google Scholar 

  22. Mas-Collel, A., Whinston, M.D., Green, J.R.: Mircoeconomic Theory. Oxford University Press, Oxford (1995)

    Google Scholar 

  23. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  24. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)

    Book  Google Scholar 

  25. Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool Publishers, San Rafael, CA (2014)

    MATH  Google Scholar 

  26. Mordukhovich, B.S., Nam, N.M.: Geometric approach to convex subdifferential calculus. Optimization 66, 839–873 (2017)

    Article  MathSciNet  Google Scholar 

  27. Mordukhovich, B.S., Nam, N.M.: Extremality of convex sets with some applications. Optim. Lett. 17, 1201–1215 (2017)

    Article  MathSciNet  Google Scholar 

  28. Mordukhovich, B.S., Nam, N.M., Rector, B., Tran, T.: Variational geometric approach to generalized differential and conjugate calculus in convex analysis. Set-Valued Var. Anal. 25, 731–755 (2017)

    Article  MathSciNet  Google Scholar 

  29. Pallaschke, D., Rolewicz, S.: Foundations of Mathematical Optimization: Convex Analysis without Linearity. Kluwer, Dordrecht (1998)

    MATH  Google Scholar 

  30. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  31. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  32. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  Google Scholar 

  33. Zălinescu, C.: On the use of the quasi-relative interior in optimization. Optimization 64, 1795–1823 (2015)

    Article  MathSciNet  Google Scholar 

  34. Zarantonello, E.H.: Projections of convex sets in Hilbert space and spectral theory, In: Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic Press, Cambridge (1971)

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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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