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Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

Abstract

This paper deals with a vector polynomial optimization problem over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called tangency varieties; obtain the relationships of the Palais–Smale condition, Cerami condition, M-tameness, and properness related to the considered problem, in which the condition of Mangasarian–Fromovitz constraint qualification at infinity plays an essential role in deriving these relationships. At last, according to the obtained connections, we establish the existence of Pareto solutions to the problem in consideration and give some examples to illustrate our main findings.

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Acknowledgements

The authors wish to thank Professor Tien-Son Pham for many valuable suggestions. We also appreciate the handling editor and referees very much for their careful comments and valuable suggestions. This work was supported by the National Natural Sciences Foundation of China (11971339, 11771319) and by Natural Science Foundation of Jilin Province (YDZJ202201ZYTS302).

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Correspondence to Pengcheng Wu.

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Communicated by Xiaoqi Yang.

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Duan, Y., Jiao, L., Wu, P. et al. Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints. J Optim Theory Appl 195, 148–171 (2022). https://doi.org/10.1007/s10957-022-02068-1

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  • DOI: https://doi.org/10.1007/s10957-022-02068-1

Keywords

  • Vector optimization
  • Polynomial optimization
  • Pareto solutions
  • Palais–Smale condition
  • Cerami condition
  • Properness

Mathematics Subject Classification

  • 90C23
  • 90C29
  • 49J30