Abstract
Sufficient conditions are given for the existence of solutions of bifunction-set optimization problems, whose underlying sets are complete metric spaces. The main result of this paper is formulated in Theorem 3.1, with the help of some new data. The advantage of introducing the new data is that, by choosing suitable new data, we can obtain existence results with assumptions, that are imposed directly on the objective maps, or that are formulated via scalarization. The main result is applied successfully to Kuroiwa set optimization problems and Ky Fan vector inequality problems. We also discuss equivalences between the existence of solutions of the bifunction-set optimization problem, an Ekeland-type variational principle and a Caristi-type fixed point theorem. Examples are provided.
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The authors would like to thank the associate editor and the anonymous referees for their valuable suggestions that improved the paper.
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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.09.
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by PHS and LAT. The first draft of the manuscript was written by PHS and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Communicated by Sándor Zoltán Németh.
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Sach, P.H., Tuan, L.A. Existence of Solutions of Bifunction-Set Optimization Problems in Metric Spaces. J Optim Theory Appl 192, 195–225 (2022). https://doi.org/10.1007/s10957-021-01958-0
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DOI: https://doi.org/10.1007/s10957-021-01958-0
Keywords
- Bifunction-set optimization
- Set optimization
- Ky Fan inequality
- Set-valued map
- Solution existence
- Metric space