Superlinear convergence of Broyden's boundedθ-class of methods

In this paper the so-called Broyden's bounded-class of methods is considered. It contains as a subclass Broyden's restricted-class of methods, in which the updating matrices retain symmetry and positive definiteness. These iteration methods are used for solving unconstrained minimization problems of the following form: It is assumed that the step-size coefficient _k = 1 in each iteration and the functionalf : R ⁿ R¹ satisfies the standard assumptions, viz.f is twice continuously differentiable and the Hessian matrix is uniformly positive definite and bounded (there exist constantsm, M > 0 such that my² y, for ally R ⁿ) and satisfies a Lipschitz-like condition at the optimal point , the gradient vanishes at Under these assumptions the local convergence of Broyden's methods is proved. Furthermore, the Q-superlinear convergence is shown.     

On the convergence of genetic algorithms – a variational approach

A variational approach is introduced to study the existence and uniqueness of stationary states and (exponential) stability of genetic algorithms with mutation and interactive selection.Mathematics Subject Classification (2000): 35J20 (90C30, 92D25, 35J60, 31C25)     

On the rate of convergence to the Bose–Einstein distribution

For a system of identical Bose particles sitting at integer energy levels with the probabilities of microstates given by a multiplicative measure with ≥ 2 degrees of freedom, we estimate the probability of the sequence of occupation numbers to be close to the Bose–Einstein distribution as the total energy tends to infinity. We show that a convergence result earlier proved by A.M. Vershik [Functional Anal. Appl. 30 (2), 95–105 (1996)] is a corollary of our theorems.     

On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem

We study the rate of convergence in von Neumann’s ergodic theorem. We obtain constants connecting the power rate of convergence of ergodic means and the power singularity at zero of the spectral measure of the corresponding dynamical system (these concepts are equivalent to each other). All the results of the paper have obvious exact analogs for wide-sense stationary stochastic processes.     

On the convergence rate of a parallel nonoverlapping domain decomposition method

In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and analyzed.It is known that the convergence rate of this method is 1-O(h),where h is mesh size.In this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain size.Counter examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.     

On the influence of the potential on the convergence rate of expansions in root functions of the Shrödinger operator

We consider the one-dimensional Schrödinger operator with integrable potential. We analyze the rate of the uniform equiconvergence of the biorthogonal expansion of an absolutely continuous function in the root functions of this operator with its Fourier trigonometric series on a compact set. For this convergence rate, we obtain an estimate depending on the modulus of continuity of the potential. We extract subclasses of absolutely continuous functions on which these estimates can be improved.     

On the rate of convergence for multi-category classification based on convex losses

The multi-category classification algorithms play an important role in both theory and practice of machine learning.In this paper,we consider an approach to the multi-category classification based on minimizing a convex surrogate of the nonstandard misclassification loss.We bound the excess misclassification error by the excess convex risk.We construct an adaptive procedure to search the classifier and furthermore obtain its convergence rate to the Bayes rule.     

On the rate of convergence for infinite server Erlang–Sevastyanov’s problem

Polynomial convergence rates in total variation are established in Erlang–Sevastyanov type problems with an infinite number of servers and a general distribution of service under assumptions on the intensity of service.     

On the convergence of “Threshold Accepting”

Simulated Annealing (SA) has become a very popular tool in combinatorial optimization since its introduction in 1982. Recently Dueck and Scheuer proposed another simple modification of local search which they called Threshold Accepting (TA). In this paper some convergence results for TA are presented. The proofs are not constructive and make use of the fact that in a certain sense SA belongs to the convex hull of TA.     

On the convergence of generalized Padé approximants

Letf be an analytic function with all its singularities in a compact set \(E_f \subset \bar C\) of (logarithmic) capacity zero. The function may have branch points. The convergence of generalized (multipoint) Padé approximants to this type of function is investigated. For appropriately selected schemes of interpolation points, it is shown that close-to-diagonal sequences of generalized Padé approximants converge in capacity tof in a certain domain that can be characterized by the property of the minimal condenser capacity. Using a pole elimination procedure, another set of rational approximants tof is derived from the considered generalized Padé approximants. These new approximants converge uniformly on a given continuum \(V \subset \bar C\backslash E_f\) with a rate of convergence that has been conjectured to be best possible. The continuumV is assumed not to divide the complex plane.     

On the O(1/t) convergence rate of the LQP prediction–correction method

The logarithmic-quadratic proximal (abbreviated by LQP) prediction–correction method is attractive for structured monotone variational inequalities, and it ensures the global convergence under some suitable conditions. In this paper, we are interested in investigating the convergence rate of the LQP prediction–correction method. Motivated by the research work about the convergence rate or iteration complexity for various first-order algorithms in the literature, we provide a simple proof to show the $O(1/t)$ convergence rate for the LQP prediction–correction method.     

A new proof of the monotone convergence theorem of Lebesgue integral on σ-class

In this short note a new proof of the monotone convergence theorem of Lebesgue integral on σ-class is given.     

On the local convergence of the Douglas–Rachford algorithm

We discuss the Douglas–Rachford algorithm to solve the feasibility problem for two closed sets A,B in \({\mathbb{R}^d}\) . We prove its local convergence to a fixed point when A,B are finite unions of convex sets. We also show that for more general nonconvex sets the scheme may fail to converge and start to cycle, and may then even fail to solve the feasibility problem.     

On the convergence of the multivariate “homogeneous”qd-algorithm

The convergence of columns in the univariateqd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariateqd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Padé approximants.In the first section we repeat the univariate convergence results. In the second section we summarize the homogeneous multivariateqd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariateqdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities pointwise while the general form detects them curvewise.     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).