{\L}ojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

TL;DR

This paper establishes explicit exponent versions of Łojasiewicz inequalities for the largest eigenvalue function of real symmetric polynomial matrices, providing new nonsmooth analysis tools and error bounds.

Contribution

It introduces a nonsmooth Łojasiewicz gradient inequality with explicit exponents for the largest eigenvalue function of polynomial matrices.

Findings

Derived explicit Łojasiewicz exponents for eigenvalue functions.

Established local and global error bounds with explicit exponents.

Extended nonsmooth analysis techniques to matrix eigenvalue functions.

Abstract

Let $F(x) := (f_{ij}(x))_{i,j=1,\ldots,p},$ be a real symmetric polynomial matrix of order $p$ and let $f(x)$ be the largest eigenvalue function of the matrix $F(x).$ We denote by ${\partial}^\circ f(x)$ the Clarke subdifferential of $f$ at $x.$ In this paper, we first give the following {\em nonsmooth} version of \L ojasiewicz gradient inequality for the function $f$ with an explicit exponent: For any $\bar x\in \Bbb R^n$ there exist $c > 0$ and $\epsilon > 0$ such that we have for all $\|x - \bar{x}\| < \epsilon,$ \begin{equation*} \inf \{ \| w \| \ : \ w \in {\partial}^\circ f(x) \} \ \ge \ c\, |f(x) - f(\bar x)|^{1 - \frac{1}{\mathscr{R}(2n+p(n+1),d+3)}}, \end{equation*} where $d:=\max_{i,j = 1, \ldots, p}\deg f_{i j}$ and $\mathscr{R}$ is a function introduced by D'Acunto and Kurdyka: $\mathscr{R}(n, d) := d(3d - 3)^{n-1}$ if $d \ge 2$ and $\mathscr{R}(n, d) := 1$ if $d = 1.$ Then we establish error bounds with explicitly determined exponents, local and global, for the largest eigenvalue function $f(x)$ of the matrix $F(x)$.