Existence of efficient and properly efficient solutions to problems of constrained vector optimization

Abstract

The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint set. The main attention is paid to the two major notions of optimality in vector problems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequate tools of variational analysis and generalized differentiation, we first establish relationships between the notions of properness, M-tameness, and the Palais–Smale conditions formulated for the restriction of the vector cost mapping on the constraint set. These results are instrumental to derive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutions in vector optimization. Furthermore, the developed approach allows us to obtain new sufficient conditions for the existence of Geoffrion-properly efficient solutions to such constrained vector problems.

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Acknowledgements

The authors are grateful to three anonymous referees and handling editors for careful reading of the original submission and for helpful suggestions and kind remarks.

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Correspondence to Nguyen Van Tuyen.

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The first author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1A2C1008672). Research of second author was supported by the US National Science Foundation under Grants DMS-1512846 and DMS-1808978, by the Air Force Office of Scientific Research under Grant #15RT0462, and by the Australian Research Council under Discovery Project DP-190100555. The third author is partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED), Grant 101.04-2019.302. The fourth author was supported by Hanoi Pedagogical University 2 (HPU2) under Grant No. C.2019-18-05.

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Kim, D.S., Mordukhovich, B.S., Phạm, T. et al. Existence of efficient and properly efficient solutions to problems of constrained vector optimization. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01532-y

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Keywords

  • Existence theorems
  • Pareto efficient solutions
  • Geoffrion-properly efficient solutions
  • M-tameness
  • Palais–Smale conditions
  • Properness

Mathematics Subject Classification

  • 90C29
  • 90C30
  • 90C31
  • 49J30