Łojasiewicz inequalities with explicit exponent for smallest singular value functions

2016/04/11 by Sĩ Tiệp Đinh, Dinh, Si Tiep, Tiến-Sơn Phạm +1 · 2 citations
Mathematics · #Advanced Banach Space Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.1604.02805

openalex publication_date 2016/04/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let F(x) := (f_ij(x))_{i=1,…,p; j=1,…,q}, be a (p× q)-real polynomial matrix and let f(x) be the smallest singular value function of F(x). In this paper, we first give the following \em nonsmooth version of Łojasiewicz gradient inequality for the function f with an explicit exponent: \em For any x∈ \Bbb Rⁿ, there exist c > 0 and ε> 0 such that we have for all ‖x - x‖ < ε, inf \ ‖ w ‖ : w ∈ ∂ f(x) \ ≥ c |f(x)-f( x)|^{1 - (2)/(\mathscr R(n+p,2d+2))}, where ∂ f(x) is the limiting subdifferential of f at x, d:=max_{i=1,…,p; j=1,…,q}°f_{i j} and \mathscr R(n, d) := d(3d - 3)^n-1 if d ≥ 2 and \mathscr R(n, d) := 1 if d = 1. Then we establish some versions of Łojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function f(x) of the matrix F(x).

Citations

• Geometric categories and o-minimal structures

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• On Error Bounds for Rank-Constrained Affine Matrix Sets
• Compact R-Continuity with Applications to Solving Inclusions and Convergence of Algorithms

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