Related papers: Lojasiewicz inequalities in a certain class of smo…

We consider the exponent of \L ojasiewicz inequality $\|\partial\,f(\mathbf z)\| \ge c |f(\mathbf z|^\theta$ for two classes of analytic functions and we will give an explicit estimation for $\theta$. First we consider certain…

The Polyak-{\L}ojasiewicz (P{\L}) condition is often invoked in nonconvex optimization because it allows fast convergence of algorithms beyond strong convexity. A function $f \colon \mathcal{M} \to \mathbb{R}$ on a Riemannian manifold…

Let $f$ be an isolated singularity at the origin of $\mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {\L}ojasiewicz exponent $\mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for…

We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in…

Let $f: (\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ be a semiquasihomogeneous function. We give a formula for the local {\L}ojasiewicz exponent $\mathcal{L}_{0}(f)$ of $f$, in terms of weights of $f$. In particular, in the case of a…

The Polyak-{\L}ojasiewicz (P{\L}) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in…

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…

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…

We give an algorithm to compute the L\^e numbers of (the germ of) a Newton non-degenerate complex analytic function $f\colon(\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ in terms of certain invariants attached to the Newton diagram of the…

The \emph{Noetherian class} is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A…

In the article we give some estimations of the {\L}ojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the {\L}ojasiewicz exponent of such singularities in some…

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems:…

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at…

We present a global version of the {\L}ojasiewicz inequality on comparing the rate of growth of two polynomial functions in the case the mapping defined by these functions is (Newton) non-degenerate at infinity. In addition, we show that…

Let $X\subset \mathbb{R}^n$ be a compact semialgebraic set and let $f:X\to \mathbb{R}$ be a nonzero Nash function. We give a Solern\'o and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the {\L}ojasiewicz gradient…

We devise calculus rules for the Kurdyka-\L{}ojasiewicz exponent using the rank theorem and Lie group actions. They apply to a wide class of composite and invariant functions, and are particularly suitable for handling nonisolated local…

Given a hypersurface singularity $(X,0) \subset (\mathbb{C}^{n+1},0)$ defined by a holomorphic function $f:(\mathbb{C}^{n+1},0) \to (\mathbb{C},0)$, we introduce an alternating version of Teissier's Jacobian Newton polygon, associated with…

This paper focuses on a class of zero-norm composite optimization problems. For this class of nonconvex nonsmooth problems, we establish the Kurdyka-Lojasiewicz property of exponent being a half for its objective function under a suitable…

For displacement convex functionals in the probability space equip\-ped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type \L oja\-sie\-wicz inequalities. \chg{We also discuss the more…

According to the Kouchnirenko theorem, for a generic (precisely non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $\mu (f)$ is equal to the Newton number $\nu (\Gamma_{+}(f))$ of a combinatorial object…