Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

A compact set K ì \mathbbC^N{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that

Convergence of Non-smooth Descent Methods Using the Kurdyka–Łojasiewicz Inequality

We investigate the convergence of subgradient-oriented descent methods in non-smooth non-convex optimization. We prove convergence in the sense of subsequences for functions with a strict standard model, and we show that convergence to a single critical point may be guaranteed if the Kurdyka–?ojasiewicz inequality is satisfied. We show, by way of an example, that the Kurdyka–?ojasiewicz inequality alone is not sufficient to prove the convergence to critical points.     

A Condition for the Existence of a Unique Green–Samoilenko Function for the Problem of Invariant Torus

Under the assumption that a linear homogeneous system defined on the direct product of a torus and a Euclidean space is exponentially dichotomous on the semiaxes, we obtain a condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus. We find an expression for this function in terms of projectors that determine the dichotomy on the semiaxes.     

Łojasiewicz–type inequalities in complex analysis

We show how some variants of the ?ojasiewicz inequality, which is a powerful tool in real analysis, can also be used to study certain problems in complex analysis and approximation theory. In particular, we discuss whether the so-called ?ojasiewicz–Siciak condition of the Siciak extremal function is preserved when taking holomorphic preimages or images.     

Multiplicities of Pfaffian intersections,and the Łojasiewicz inequality

An effective estimate for the local multiplicity of a complete intersection of complex algebraic and Pfaffian varieties is given, based on a local complex analog of the Rolle-Khovanskii theorem. The estimate is valid also for the properly defined multiplicity of a non-isolated intersection. It implies, in particular, effective estimates for the exponents of the polar curves, and the exponents in the ojasiewicz inequalities for Pfaffian functions. For the intersections defined by sparse polynomials, the multiplicities outside the coordinate hyperplanes can be estimated in terms of the number of non-zero monomials, independent of degrees of the monomials.     

Łojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of O_X{\mathcal{O}_X} , where X is a non-singular scheme. This method is based on the algorithm of resolution of singularities.     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings

We prove a sharp regular separation condition for arbitrary projective algebraic sets. As a corollary an extension of Kollár's results on the Łojasiewicz exponent of polynomial mappings is obtained. Oblatum 9-II-1998 & 3-VI-1998 / Published online: 14 January 1999     

Correction to: Kurdyka–Łojasiewicz Property of Zero-Norm Composite Functions

Journal of Optimization Theory and Applications - A correction to this paper has been published: https://doi.org/10.1007/s10957-020-01779-7.     

Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods

In this paper, we study the Kurdyka–?ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is \(\frac{1}{2}\). The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is \(\frac{1}{2}\). This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with \(\ell _1\) regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery.     

Łojasiewicz inequalities in o-minimal structures

This note is devoted to the generalization of ?ojasiewicz inequalities for functions definable in o-minimal structures, which is, roughly speaking, a generalization for semialgebraic or global subanalytic functions. We present some o-minimal versions of the inequalities to compare two definable functions globally or in some neighborhoods of the zero-sets of the functions, and the gradient inequalities (Kurdyka–?ojasiewicz inequality and Bochnak–?ojasiewicz inequality). Some applications of the inequalities are given.     

On The Effective Computation Of Łojasiewicz Exponents Via Newton Polyhedra

In this paper we give some conclusions on Newton non-degenerate analytic map germs on Kn (K = ? or ?), using information from their Newton polyhedra. As a consequence, we obtain the exact value of the Lojasiewicz exponent at the origin of Newton non-degenerate analytic map germs. In particular, we establish a connection between Newton non-degenerate ideals and their integral closures, thus leading to a simple proof of a result of Saia. Similar results are also considered to polynomial maps which are Newton non-degenerate at infinity.     

Injectivity of real polynomial maps and Łojasiewicz exponents at infinity

In this paper we give a lower bound for the Łojasiewicz exponent at infinity of a special class of polynomial maps , s ≥ 1. As a consequence, we detect a class of polynomial maps that are global diffeomorphisms if their Jacobian determinant never vanishes. Work supported by DGICYT Grant BFM2003–02037/MATE.     

Local Łojasiewicz exponents,Milnor numbers and mixed multiplicities of ideals

Let be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I ₁,..., I _n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions. Work supported by DGICYT Grant MTM2006-06027.     

The ?ojasiewicz exponent of nondegenerate surface singularities

We give some estimations of the ?ojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the ?ojasiewicz exponent of such singularities in some special cases. The results are stronger than Fukui inequality?[8]. It is also a multidimensional generalization of the Lenarcik theorem?[13].     

The Generalized Auslander–Reiten Condition for the bounded derived category

We consider the bounded derived category D ^b (R mod) of a left Noetherian ring R. We give a version of the Generalized Auslander–Reiten Condition for D ^b (R mod) that is equivalent to the classical statement for the module category and is preserved under derived equivalences.     

Multi-block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates Under Kurdyka–Łojasiewicz Property

Journal of Optimization Theory and Applications - We study the convergence and convergence rates of a multi-block proximal alternating direction method of multipliers (PADMM) for solving linearly...     

On Smoothness of the Green Function for?the?Complement of a Rarefied Cantor-Type Set

Smoothness of the Green functions for the complement of rarefied Cantor-type sets is described in terms of the function j(d)=(1/logfrac1d)varphi (delta)=(1/logfrac{1}{delta}) that gives the logarithmic measure of sets. Markov’s constants of the corresponding sets are evaluated.