The modes of convergence in the approximation of fuzzy random optimization problems

To develop the approximation approach to fuzzy random optimization problems, it is required to introduce the modes of convergence in fuzzy random theory. For this purpose, this paper first presents several novel convergence concepts for sequences of fuzzy random variables, such as convergence in chance, convergence in distribution and convergence in optimistic value; then deals with the convergence criteria and convergence relations among various types of convergence. Finally, we deal with the convergence theorems for sequences of integrable fuzzy random variables, including dominated convergence theorem and bounded convergence theorem.     

CMAC学习过程收敛性的研究

基于CMAC学习过程等价于求解线性方程组的(Gauss-Seidel迭代这一事实,研究 了学习过程的收敛性.利用矩阵分析方法,估计出了收敛的速度.考虑了作为节省存储空间措 施的hash编码的不利影响--破坏了收敛性态.从理论上分析了其存在的原因.     

On the monotonic convergence of the eigensolution by the h-version of the finite element method

The convergence of the strain energy of solutions by the finite element method (FEM) for static problems has been extensively investigated. It has been proven that for exactly and minimally conforming hierarchical elements, convergence is monotonic as the number of degrees of freedom (NDOF) are increased by both the h- and p-versions.An investigation of the convergence of the eigensolution for dynamic problems by the h-version is presented. In general, an increase in the NDOF does not result in an improvement in the eigensolution. A condition for refinement is given under which monotonic convergence and improvement is ensured.     

Criterion for the convergence of the solution of the Riccati differential equation

The optimal control problem for a linear system with a quadratic cost function leads to the matrix Riccati differential equation. The convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix. A necessary and sufficient condition for convergence is derived for stabilizable systems, even if the output in the cost function is not detectable. An algorithm is developed to determine the limiting value of the solution, which is one of the symmetric positive semidefinite solutions of the algebraic Riccati equation. Examples for convergence and nonconvergence are given. A discussion is also included of the convergence properties of the solution of the Riccati differential equation to any real symmetric (not necessarily positive semidefinite) solution of the algebraic Riccati equation.     

鞅超收敛定理与遗忘因子最小二乘算法的收敛法分析

本文扩展了用于分析时不变系统辨识算法收敛性的鞅收敛定理，建立了鞅超收敛定理。它可以作为工具来分析时变系统的各种辨识算法的收敛性，为地变系统收敛性和稳定性分析这一困难课题提供了新方法，开辟了新路。     

On the improved Newton-like methods for the inclusion of polynomial zeros

The aim of this paper is to present some modifications of Newton's type method for the simultaneous inclusion of all simple complex zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, the convergence analysis shows that the convergence rate of the basic method is increased from 3 to 6 using Jarratt's corrections. The proposed method possesses a great computational efficiency since the acceleration of convergence is attained with only few additional calculations. Numerical results are given to demonstrate convergence properties of the considered methods.     

On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces

Recently, the concept of statistical convergence of double sequences has been studied in intuitionistic fuzzy normed spaces by Mursaleen and Mohiuddine [Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos, Solitons Fractals, 41 (2009) pp. 2414–2421]. We know that ideal convergence is more general than statistical convergence for single or double sequences. This has motivated us to study the ideal convergence of double sequences in a more general setting. That is, in this paper, we study the concept of ideal convergence and ideal Cauchy for double sequences in intuitionistic fuzzy normed spaces.     

On the convergence of the decomposition method for support vectormachines

The decomposition method is currently one of the major methods for solving support vector machines (SVM). Its convergence properties have not been fully understood. The general asymptotic convergence was first proposed by Chang et al. However, their working set selection does not coincide with existing implementation. A later breakthrough by Keerthi and Gilbert (2000, 2002) proved the convergence finite termination for practical cases while the size of the working set is restricted to two. In this paper, we prove the asymptotic convergence of the algorithm used by the software SVM(light) and other later implementation. The size of the working set can be any even number. Extensions to other SVM formulations are also discussed.     

On the almost sure and mean-square exponential convergence of somestochastic observers

It is pointed out that linear observers used for estimating the state of the discrete-time stochastic-parameter systems are both almost surely and mean-square (MS) exponentially convergent under the same conditions guaranteeing mean-square convergence. In addition to the mean-square convergence properties of linear observers constructed for mean-square stable stochastic-parameter systems, they also possess an almost-sure exponential convergence property, and the rate of MS convergence is exponential. This rate depends on the parameters used in the design     

Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front

In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent.     

Generalization of the Dvoretzky theorem of convergence rate of the stochastic approximation algorithms

Consideration was given to the Robbins–Monro procedure for which the Dvoretzky theorem of its convergence rate was generalized. The rate of convergence is still the most important problem of the theory of stochastic approximation.     

On the convergence of the discrete ART algorithm for the reconstruction of digital pictures from their projections

New Proofs are given for the known convergence of the additive and linear (i.e. unconstrained) ART algorithms. These algorithms belong to a class of methods for the reconstruction of digitized pictures from one-dimensional views which are used e. g. in x-ray tomography. Avoiding the detour of solving systems of inequalities, the first proof gives, simultaneously and in a very direct way, the convergence of both the additive and the linear algorithms. A second proof shows the geometric convergence of the linear algorithm by using elementary matrix algebra only.     

On positive real transfer functions and the convergence of some recursive schemes

The convergence with probability one of a recently suggested recursive identification method by Landau is investigated. The positive realness of a certain transfer function is shown to play a crucial role, both for the proof of convergence and for convergence itself. A completely analogous analysis can be performed also for the extended least squares method and for the self-tuning regulator of Åström and Wittenmark. Explicit conditions for convergence of all these schemes are given. A more general structure is also discussed, as well as relations to other recursive algorithms.     

Characterization of the convergence of stationary Fokker–Planck learning

The convergence properties of the stationary Fokker–Planck algorithm for the estimation of the asymptotic density of stochastic search processes is studied. Theoretical and empirical arguments for the characterization of convergence of the estimation in the case of separable and nonseparable nonlinear optimization problems are given. Some implications of the convergence of stationary Fokker–Planck learning for the inference of parameters in artificial neural network models are outlined.     

Analysis of the convergence properties of topology preservingneural networks

The authors provide a rigorous treatment of the convergence of the topology preserving neural networks proposed by Kohonen for the one-dimensional case. The approach extends the original work by Kohonen on the convergence properties of such networks in several respects. First, the authors investigate the convergence of the neuron weights directly as compared to Kohonen's treatment of the dynamic behavior of the expectation values of the weights. Second, the problem is formulated for a more general case of selecting the neighborhood amplitude of interaction rather than the uniform amplitude. Third, the proof of convergence is based on the well-known Gladyshev theorem which uses Lyapunov's function method. The authors provide a step-by-step constructive proof which establishes the asymptotic convergence to a unique solution. This proof also provides the relation between the boundary neurons' weight vectors and the number of neurons in the network. The approach is then extended to the two-dimensional case and the result is stated in a theorem.     

On the convergence of subproper (multi)-splitting methods for solving rectangular linear systems

Abstract We give a convergence criterion for stationary iterative schemes based on subproper splittings for solving rectangular systems and show that, for special splittings, convergence and quotient convergence are equivalent. We also analyze the convergence of multisplitting algorithms for the solution of rectangular systems when the coefficient matrices have special properties and the linear systems are consistent. Keywords: Rectangular linear system, iterative method, proper splitting, subproper splitting, regularity, Hermitian positive semi-definite matrix, multi-splitting, quotient convergence AMS Subject Classification: 65F10, 65F15     

The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros

In this paper, we give the convergence analysis of the Euler-like iterative method for the simultaneous inclusion of all simple real or complex zeros of a polynomial. The established initial conditions provide the safe convergence of the considered method and the fourth order of convergence. These conditions are computationally verifiable, which is of practical importance. A procedure for the choice of initial inclusion disks is also given.     

Asymptotic convergence properties of the extended Kalman filter using filtered state estimates

In a recent paper, Ljung has given a convergence analysis of the extended Kalman filter (EKF) as a parameter estimator for linear systems. The analysis is done for a version of the EKF using predicted values of the state vector. In this note a similar convergence analysis is done for the EKF using filtered values of the state vector. The convergence properties of the two algorithms are similar, but not identical. The recalculation of a simple example given by Ljung indicates that using the filtered estimate of the state vector gives improved convergence properties of the algorithm.     

On the convergence condition and convergence time of BGP

The Internet composes of thousands of Autonomous System (ASes). The Border Gateway Protocol (BGP) is the standard protocol for sharing inter-domain routing information. Unlike OSPF and IS-IS, BGP allows an AS to use a lot of attributes to express semantic rich routing policies that are consistent with its desired economic, business, performance, and security goals. However, the expressiveness could cause to delay convergence or even divergence in BGP. Recent work do not rigorously analyze the impact of the general routing policies on the convergence condition and convergence time of BGP, especially considering the widely used Multi-Exit Discriminator (MED) attribute. In this paper, we will fill this gap and give the rigorous analysis on the impact of the general routing policies on the convergence condition and convergence time of BGP, including MED attribute. We first introduce a timeless model to represent BGP with the general routing policies including the MED attribute. By incorporating the timeless model we derive a sufficient condition on these general routing policies for robust convergence of BGP. We then extend the timeless model to the real-time model by adding the edge delay. Finally, we find an upper bound on convergence time of BGP by incorporating the real-time model.     

Convergence of the iterative Hammerstein system identification algorithm

The convergence of the iterative identification algorithm for the Hammerstein system has been an open problem for a long time. In this paper, a detailed study is carried out and various convergence properties of the iterative algorithm are derived. It is shown that the iterative algorithm with normalization is convergent in general. Moreover, it is shown that convergence takes place in one step (two least squares iterations) for finite-impulse response Hammerstein models with i.i.d. inputs.