On the existence of Pareto solutions for polynomial vector optimization problems

  • Do Sang Kim
  • Tiến-Sơn Phạm
  • Nguyen Van Tuyen
Full Length Paper Series A

Abstract

We are interested in the existence of Pareto solutions to the vector optimization problem
$$\begin{aligned} \mathrm{Min}_{\,{\mathbb {R}}^m_+} \{f(x) \,|\, x\in {\mathbb {R}}^n\}, \end{aligned}$$
where \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^m\) is a polynomial map. By using the tangency variety of f we first construct a semi-algebraic set of dimension at most \(m - 1\) containing the set of Pareto values of the problem. Then we establish connections between the Palais–Smale conditions, M-tameness, and properness for the map f. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions of the problem. We also introduce a generic class of polynomial vector optimization problems having at least one Pareto solution.

Keywords

Existence theorems Pareto solutions M-tameness Palais–Smale conditions Properness Polynomial 

Mathematics Subject Classification

90C29 90C30 49J30 

Notes

Acknowledgements

The authors would like to thank the three referees for careful reading and constructive comments. A part of this work was done while the second and third authors were visiting Department of Applied Mathematics, Pukyong National University, Busan, Korea in September 2016. These authors would like to thank the department for hospitality and support during their stay.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  • Do Sang Kim
    • 1
  • Tiến-Sơn Phạm
    • 2
  • Nguyen Van Tuyen
    • 3
  1. 1.Department of Applied MathematicsPukyong National UniversityBusanKorea
  2. 2.Department of MathematicsUniversity of DalatDa LatVietnam
  3. 3.Department of MathematicsHanoi Pedagogical University 2Xuan Hoa, Phuc YenVietnam

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