{\L}ojasiewicz inequalities with explicit exponent for smallest singular value functions

TL;DR

This paper establishes explicit exponent versions of the Łojasiewicz inequality for the smallest singular value function of polynomial matrices, providing new nonsmooth analysis tools with potential applications in optimization and stability analysis.

Contribution

It introduces a nonsmooth Łojasiewicz gradient inequality with explicit exponents for the smallest singular value function of polynomial matrices, extending classical results.

Findings

Derived explicit Łojasiewicz exponents for singular value functions

Established local and global inequalities for distance functions

Provided tools for stability and optimization analysis involving polynomial matrices

Abstract

Let $F(x) := (f_{ij}(x))_{i=1,\ldots,p; j=1,\ldots,q},$ 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 \L ojasiewicz gradient inequality for the function $f$ with an explicit exponent: {\em 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} 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 $x$, $d:=\max_{i=1,\ldots,p; j=1,\ldots,q}\deg f_{i j}$ and $\mathscr R(n, d) := d(3d - 3)^{n-1}$ if $d \ge 2$ and $\mathscr R(n, d) := 1$ if $d = 1.$} Then we establish some versions of \L ojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function $f(x)$ of the matrix $F(x)$.