Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 563–585

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

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Abstract

Let F := (f1, …, fp): ℝn → ℝp be a polynomial map, and suppose that S := {x ∈ ℝn : fi(x) ≤ 0,i = 1, …, p}≠. Let d := maxi =1, …, p deg fi 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, …, pfi(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).

Keywords

Error bounds Newton polyhedron Non-degenerate polynomial maps Palais-Smale condition 

Mathematics Subject Classification (2010)

Primary 32B20 Secondary 14P 49K40 

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of MathematicsUniversity of DalatDa LatVietnam

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