On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

Weak Convergence of n-Particle Systems Using Bilinear Forms

The paper is concerned with the weak convergence of n-particle processes to deterministic stationary paths as ${n \rightarrow \infty}$ . A Mosco type convergence of a class of bilinear forms is introduced. The Mosco type convergence of bilinear forms results in a certain convergence of the resolvents of the n-particle systems. Based on this convergence a criterion in order to verify weak convergence of invariant measures is established. Under additional conditions weak convergence of stationary n-particle processes to stationary deterministic paths is proved. The method is applied to the particle approximation of a Ginzburg-Landau type diffusion. The present paper is in close relation to the paper [9]. Different definitions of bilinear forms and versions of Mosco type convergence are introduced. Both papers demonstrate that the choice of the form and the type of convergence relates to the particular particle system.     

Convergence of the Nelder-Mead method

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

     

Convergence rate of McCormick relaxations

Theory for the convergence order of the convex relaxations by McCormick (Math Program 10(1):147–175, 1976) for factorable functions is developed. Convergence rules are established for the addition, multiplication and composition operations. The convergence order is considered both in terms of pointwise convergence and of convergence in the Hausdorff metric. The convergence order of the composite function depends on the convergence order of the relaxations of the factors. No improvement in the order of convergence compared to that of the underlying bound calculation, e.g., via interval extensions, can be guaranteed unless the relaxations of the factors have pointwise convergence of high order. The McCormick relaxations are compared with the αBB relaxations by Floudas and coworkers (J Chem Phys, 1992, J Glob Optim, 1995, 1996), which guarantee quadratic convergence. Illustrative and numerical examples are given.     

Analyzing the convergence factor of residual inverse iteration

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w _k and we can use the formula for the convergence factor to analyze how it depends on the choice of w _k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.     

Local properties of algorithms for minimizing nonsmooth composite functions

This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems.This work was done while the author was a Research fellow at the Mathematical Sciences Research Centre, Australian National University.     

The convergence space of minimal USCO mappings

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all continuous functions is also addressed.      

Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations

Necessary and sufficient conditions under which two- and three-point iterative methods have the order of convergence р (2 ≤ р ≤ 8) are formulated for the first time. These conditions can be effectively used to prove the convergence of iterative methods. In particular, the order of convergence of some known optimal methods is verified using the proposed sufficient convergence tests. The optimal set of parameters making it possible to increase the order of convergence is found. It is shown that the parameters of the known iterative methods with the optimal order of convergence have the same asymptotic behavior. The simplicity of choosing the parameters of the proposed methods is an advantage over the other known methods.     

Ideas from Continued Fraction Theory Extended to Padé Approximation and Generalized Iteration

This is a survey of some basic ideas in the convergence theory for continued fractions, in particular value sets, general convergence and the use of modified approximants to obtain convergence acceleration and analytic continuation. The purpose is to show how these ideas apply to some other areas of mathematics. In particular, we introduce {w _k }-modifications and general convergence for sequences of Padé approximants.     

Lim-inf convergence in partially ordered sets

The lim-inf convergence in a complete lattice was introduced by Scott to characterize continuous lattices. Here we introduce and study the lim-inf convergence in a partially ordered set. The main result is that for a poset P the lim-inf convergence is topological if and only if P is a continuous poset. A weaker form of lim-inf convergence in posets is also discussed.     

Local convergence in Fermat's problem

The general Fermat problem is to find the minimum of the weighted sum of distances fromm destination points in Euclideann-space. Kuhn recently proved that a classical iterative algorithm converges to the unique minimizing point , for any choice of the initial point except for a denumerable set. In this note, it is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination. If  is not a destination, then locally convergence is always linear with upper and lower asymptotic convergence boundsλ andλ′ (λ ≥ 1/2, whenn=2). If  is a destination, then convergence can be either linear, quadratic or sublinear. Three numerical examples which illustrate the different possibilities are given and comparisons are made with the use of Steffensen's scheme to accelerate convergence.     

Nonlinear approximation by renormalized trigonometric system

We study the convergence of greedy algorithmwith regard to renormalized trigonometric system. Necessary and sufficient conditions are found for system’s normalization to guarantee almost everywhere convergence, and convergence in L ^p (T) for 1 < p < ∞ of the greedy algorithm, where T is the unit torus. Also the non existence is proved for normalization which guarantees convergence almost everywhere for functions from L ¹(T), or uniform convergence for continuous functions.     

Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.     

Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.      

Convergence of Polynomial Level Sets

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.

     

Local convergence theorems for Newton methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on themth (m ≥ 2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Fréchet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

The quasi-metric of complexity convergence

For any weightable quasi-metric space (X, d) having a maximum with respect to the associated order ≤ _d , the notion of the quasi-metric of complexity convergence on the the function space (equivalently, the space of sequences) X^ω , is introduced and studied. We observe that its induced quasi-uniformity is finer than the quasi-uniformity of pointwise convergence and weaker than the quasi-uniformity of uniform convergence. We show that it coincides with the quasi-uniformity of pointwise convergence if and only if the quasi-metric space (X, d) is bounded and it coincides with the quasi-uniformity of uniform convergence if and only if X is a singleton. We also investigate completeness of the quasi-metric of complexity convergence. Finally, we obtain versions of the celebrated Grothendieck theorem in this context.     

Affine invariant local convergence theorems for inexact Newton-like methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Fréchet-derivative whereas the second theorem employs hypotheses on the second. Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the second Fréchet-derivative our radius of convergence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].     

Slow convergence of sequences of linear operators II: Arbitrarily slow convergence

We study the rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. Our main interest is in characterizing the slowest type of pointwise convergence possible. This is a continuation of the paper Deutsch and Hundal (2010) [14]. The main result is a “lethargy” theorem (Theorem 3.3) which gives useful conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, we obtain a “dichotomy” theorem, which states the surprising result that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. The dichotomy theorem is applied to generalize and sharpen: (1) the von Neumann–Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov.