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On the solution existence and stability of polynomial optimization problems

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Abstract

In this paper, we introduce and investigate a new regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. The normalization argument in asymptotic analysis enables us to study the existence as well as the stability of solutions of these problems. We prove a Frank–Wolfe type theorem for regular optimization problems and an Eaves type theorem for non-regular pseudoconvex optimization problems. Moreover, under the regularity condition, we show results on the stability such as upper semicontinuity and local upper-Hölder stability of the solution map of polynomial optimization problems. At the end of the paper, we discuss the genericity of the regularity condition.

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Acknowledgements

The author would like to thank the anonymous referees for their corrections and comments. This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Actions, grant agreement 813211 (POEMA).

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Authors and Affiliations

  1. CNRS, LIP6, Sorbonne Université, 75005, Paris, France

    Vu Trung Hieu

  2. Division of Mathematics, Phuong Dong University, 171 Trung Kinh Street, Hanoi, Vietnam

    Vu Trung Hieu

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  1. Vu Trung Hieu
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Correspondence to Vu Trung Hieu.

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Hieu, V.T. On the solution existence and stability of polynomial optimization problems. Optim Lett 16, 1513–1529 (2022). https://doi.org/10.1007/s11590-021-01788-z

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  • DOI: https://doi.org/10.1007/s11590-021-01788-z

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