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

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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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Authors and Affiliations

  1. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Republic of Korea

    Do Sang Kim

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

    Boris S. Mordukhovich

  3. RUDN University, Moscow, Russia, 117198

    Boris S. Mordukhovich

  4. Department of Mathematics, University of Dalat, 1 Phu Dong Thien Vuong, Dalat, Vietnam

    Tiến-Sơn Phạm

  5. Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam

    Nguyen Van Tuyen

  6. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, People’s Republic of China

    Nguyen Van Tuyen

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  1. Do Sang Kim
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  2. Boris S. Mordukhovich
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  3. Tiến-Sơn Phạm
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  4. Nguyen Van Tuyen
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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, TS. et al. Existence of efficient and properly efficient solutions to problems of constrained vector optimization. Math. Program. 190, 259–283 (2021). https://doi.org/10.1007/s10107-020-01532-y

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  • DOI: https://doi.org/10.1007/s10107-020-01532-y

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