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A Frank–Wolfe type theorem for nondegenerate polynomial programs

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Abstract

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let \(f_0\) and \(f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be convenient polynomial functions, and let \(S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .\) Under the assumption that the map \((f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}\) is non-degenerate at infinity, we show that if \(f_0\) is bounded from below on \(S,\) then \(f_0\) attains its infimum on \(S.\)

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Acknowledgments

The authors would like to thank the referees, whose helpful comments and suggestions much improved the original manuscript. This research was performed while the authors had been visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the Institute for hospitality and support.

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Authors and Affiliations

  1. Institute of Mathematics, 18, Hoang Quoc Viet Road, Cau Giay District, 10307 , Hanoi, Vietnam

    Si Tiep Dinh & Huy Vui Ha

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

    Tien Son Pham

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  1. Si Tiep Dinh
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  2. Huy Vui Ha
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  3. Tien Son Pham
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Correspondence to Tien Son Pham.

Additional information

Si Tiep Dinh and Huy Vui Ha: These authors were partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) Grant Numbers 101.01-2011.44.

Tien Son Pham: This author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) Grant Numbers 101.04-2013.07.

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Dinh, S.T., Ha, H.V. & Pham, T.S. A Frank–Wolfe type theorem for nondegenerate polynomial programs. Math. Program. 147, 519–538 (2014). https://doi.org/10.1007/s10107-013-0732-2

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  • DOI: https://doi.org/10.1007/s10107-013-0732-2

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