Łojasiewicz inequalities with explicit exponent for smallest singular value functions

Abstract

Let $F(x):=(f_{ij}(x){)}_{i=1,\dots ,p;j=1,\dots ,q},$ be a ($p\times q$)-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 $\bar{x}\in {\mathbb{R}}^n$, 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} f(x) \} \ \ge \ c\, |f(x)-f(\bar x)|^{1 - \frac{2}{\mathscr R(n+p,2d+2)}}, \end{equation*} where $\partial f(x)$ is the limiting subdifferential of $f$ at $x$, $d:={\max }_{i=1,\dots ,p;j=1,\dots ,q}°f_{ij}$ and $\mathcal{R}(n,d):=d(3d-3{)}^{n-1}$ if $d\ge 2$ and $\mathcal{R}(n,d):=1$ if $d=1.$} Then we establish 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 $F(x)$.