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

2014/11/04 by Dinh, Si Tiep, Vui, Ha Huy, Son, Pham Tien · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1411.0859

Abstract

Let F := (f₁, …, f_p) \colon \Bbb Rⁿ → \Bbb R^p be a polynomial map, and suppose that S := \x ∈ \Bbb Rⁿ : f_i(x) ≤ 0, i = 1, …, p\ ≠ ∅. Let d := max_{i = 1, …, p} °f_i and H(d, n, p) := d(6d - 3)^{n + p - 1}. Under the assumption that the map F \colon \Bbb Rⁿ → \Bbb R^p is convenient and non-degenerate at infinity, we show that there exists a constant c > 0 such that the following so-called \em Hölder-type global error bound result holds c d(x,S) ≤ [f(x)]₊^{(2)/(H(2d, n, p))} + [f(x)]₊ \textrm for all x ∈ ℝⁿ, where d(x, S) denotes the Euclidean distance between x and S, f(x) := max_{i = 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.

Cited by

• Exact penalty functions and global saddle points of augmented Lagrangians for well-posed constrained optimization problems

Related

• Fedoryuk values and stability of global H "olderian error bounds for\n polynomial functions
• Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates
• Global Łojasiewicz inequalities on comparing the rate of growth of polynomial functions
• On Globally Diffeomorphic Polynomial Maps via Newton Polytopes and Circuit Numbers
• Bifurcation sets and global monodromies of Newton non-degenerate polynomials on algebraic sets