arXiv subjects

Tien Son Pham

Publications and source records attributed to Tien Son Pham.

21 records · Page 2Linked to original sources

arXiv2016

Łojasiewicz inequalities with explicit exponent for smallest singular value functions

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 Łojasiewicz gradient inequality for the function $f$ with an explicit exponent: {\em For any $\bar x\in \Bbb R^n$, there exist $c > 0$ and $ε> 0$ such that we have for all $\|x - \bar{x}\| < ε,$ \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 $x$, $d:=\max_{i=1,\ldots,p; j=1,\ldots,q}°f_{i j}$ 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 Łojasiewicz inequality for the distance function with explicit exponents, locally and globally, for the smallest singular value function $f(x)$ of the matrix $F(x)$.

math.AG↗

arXiv2016

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

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 Łojasiewicz gradient inequality for the function $f$ with an explicit exponent: For any $\bar x\in \Bbb R^n$ there exist $c > 0$ and $ε> 0$ such that we have for all $\|x - \bar{x}\| < ε,$ \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 $d:=\max_{i,j = 1, \ldots, p}°f_{i j}$ 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 $F(x)$.

math.AG↗

arXiv2015

Semidefinite approximations of the polynomial abscissa

Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is H{ö}lder continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In this paper we propose simple approximations of the abscissa given by polynomials of fixed degree, and hence controlled complexity. Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems. When their degree tends to infinity, the polynomial approximations converge in norm to the abcissa, either from above or from below.

math.OC↗