Łojasiewicz-type inequalities with explicit exponents for the largest eigenvalue function of real symmetric polynomial matrices

Abstract

Let $F(x):=(f_{ij}(x){)}_{i,j=1,\dots ,p},$ 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 Łojasiewicz gradient inequality for the function $f$ with an explicit exponent: For any $\bar{x}\in {\mathbb{R}}^n$ there exist $c>0$ and $\epsilon >0$ such that we have for all $\Vert x-\bar{x}\Vert <\epsilon ,$ \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 $d:={\max }_{i,j=1,\dots ,p}°f_{ij}$ and $\mathcal{R}$ is a function introduced by D'Acunto and Kurdyka: $\mathcal{R}(n,d):=d(3d-3{)}^{n-1}$ if $d\ge 2$ and $\mathcal{R}(n,d):=1$ 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 $F(x)$.

arXiv admin note: text overlap with arXiv:1411.0859