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Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems

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Abstract

Let F := (f 1, …, f p ): ℝn → ℝp be a polynomial map, and suppose that S := {x ∈ ℝn : f i (x) ≤ 0,i = 1, …, p}≠∅. Let d := maxi =1, …, p deg f i and \(\mathcal {H}(d, n, p) := d(6d - 3)^{n + p - 1}.\) Under the assumptions that the map F : ℝn → ℝp is convenient and non-degenerate at infinity, we show that there exists a constant c > 0 such that the following so-called Hölder-type global error bound result holds \(c d(x,S) \le [f(x)]_{+}^{\frac {2}{\mathcal {H}(2d, n, p)}} + [f(x)]_{+} \quad \textrm { for all } \quad x \in \mathbb {R}^{n},\) where d(x,S) denotes the Euclidean distance between x and S, f(x) := maxi=1, …, p f i (x), and [f(x)]+ := max{f(x),0}. The class of polynomial maps (with fixed Newton polyhedra), which are non-degenerate at infinity, is generic in the sense that it is an open and dense semi-algebraic set. Therefore, Hölder-type global error bounds hold for a large class of polynomial maps, which can be recognized relatively easily from their combinatoric data. This follows up the result on a Frank-Wolfe type theorem for non-degenerate polynomial programs in Dinh et al. (Mathematical Programming Series A, 147(16), 519–538, 2014).

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Acknowledgements

This research was performed while the authors were visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the Institute for hospitality and support. Si Tiep Dinh and Huy Vui Ha were partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.04-2014.23 and the Vietnam Academy of Science and Technology (VAST). Tien Son Pham was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.04-2016.05.

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

    Si Tiep Dinh & Huy Vui Ha

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

    Tien Son Pham

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  1. Si Tiep Dinh
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Dinh, S.T., Ha, H.V. & Pham, T.S. Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems. Acta Math Vietnam 42, 563–585 (2017). https://doi.org/10.1007/s40306-017-0209-0

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