Showing papers on "Convergence (routing) published in 2002"

A survey of convergence results on particle filtering methods for practitioners

Abstract: Optimal filtering problems are ubiquitous in signal processing and related fields. Except for a restricted class of models, the optimal filter does not admit a closed-form expression. Particle filtering methods are a set of flexible and powerful sequential Monte Carlo methods designed to. solve the optimal filtering problem numerically. The posterior distribution of the state is approximated by a large set of Dirac-delta masses (samples/particles) that evolve randomly in time according to the dynamics of the model and the observations. The particles are interacting; thus, classical limit theorems relying on statistically independent samples do not apply. In this paper, our aim is to present a survey of convergence results on this class of methods to make them accessible to practitioners.

A genetic algorithm for shortest path routing problem and the sizing of populations

Abstract: This paper presents a genetic algorithmic approach to the shortest path (SP) routing problem. Variable-length chromosomes (strings) and their genes (parameters) have been used for encoding the problem. The crossover operation exchanges partial chromosomes (partial routes) at positionally independent crossing sites and the mutation operation maintains the genetic diversity of the population. The proposed algorithm can cure all the infeasible chromosomes with a simple repair function. Crossover and mutation together provide a search capability that results in improved quality of solution and enhanced rate of convergence. This paper also develops a population-sizing equation that facilitates a solution with desired quality. It is based on the gambler ruin model; the equation has been further enhanced and generalized. The equation relates the size of the population, quality of solution, cardinality of the alphabet, and other parameters of the proposed algorithm. Computer simulations show that the proposed algorithm exhibits a much better quality of solution (route optimality) and a much higher rate of convergence than other algorithms. The results are relatively independent of problem types for almost all source-destination pairs. Furthermore, simulation studies emphasize the usefulness of the population-sizing equation. The equation scales to larger networks. It is felt that it can be used for determining an adequate population size in the SP routing problem.

A framework for evolutionary optimization with approximate fitness functions

Abstract: It is not unusual that an approximate model is needed for fitness evaluation in evolutionary computation. In this case, the convergence properties of the evolutionary algorithm are unclear due to the approximation error of the model. In this paper, extensive empirical studies are carried out to investigate the convergence properties of an evolution strategy using an approximate fitness function on two benchmark problems. It is found that incorrect convergence will occur if the approximate model has false optima. To address this problem, individual- and generation-based evolution control are introduced and the resulting effects on the convergence properties are presented. A framework for managing approximate models in generation-based evolution control is proposed. This framework is well suited for parallel evolutionary optimization, which is able to guarantee the correct convergence of the evolutionary algorithm, as well as to reduce the computation cost as much as possible. Control of the evolution and updating of the approximate models are based on the estimated fidelity of the approximate model. Numerical results are presented for three test problems and for an aerodynamic design example.

Adaptive dynamic programming

Abstract: Unlike the many soft computing applications where it suffices to achieve a "good approximation most of the time," a control system must be stable all of the time. As such, if one desires to learn a control law in real-time, a fusion of soft computing techniques to learn the appropriate control law with hard computing techniques to maintain the stability constraint and guarantee convergence is required. The objective of the paper is to describe an adaptive dynamic programming algorithm (ADPA) which fuses soft computing techniques to learn the optimal cost (or return) functional for a stabilizable nonlinear system with unknown dynamics and hard computing techniques to verify the stability and convergence of the algorithm. Specifically, the algorithm is initialized with a (stabilizing) cost functional and the system is run with the corresponding control law (defined by the Hamilton-Jacobi-Bellman equation), with the resultant state trajectories used to update the cost functional in a soft computing mode. Hard computing techniques are then used to show that this process is globally convergent with stepwise stability to the optimal cost functional/control law pair for an (unknown) input affine system with an input quadratic performance measure (modulo the appropriate technical conditions). Three specific implementations of the ADPA are developed for 1) the linear case, 2) for the nonlinear case using a locally quadratic approximation to the cost functional, and 3) the nonlinear case using a radial basis function approximation of the cost functional; illustrated by applications to flight control.

Characterizing the Internet hierarchy from multiple vantage points

Abstract: The delivery of IP traffic through the Internet depends on the complex interactions between thousands of autonomous systems (AS) that exchange routing information using the border gateway protocol (BGP). This paper investigates the topological structure of the Internet in terms of customer-provider and peer-peer relationships between autonomous systems, as manifested in BGP routing policies. We describe a technique for inferring AS relationships by exploiting partial views of the AS graph available from different vantage points. Next we apply the technique to a collection of ten BGP routing tables to infer the relationships between neighboring autonomous systems. Based on these results, we analyze the hierarchical structure of the Internet and propose a five-level classification of AS. Our characterization differs from previous studies by focusing on the commercial relationships between autonomous systems rather than simply the connectivity between the nodes.

How can extremism prevail? A study based on the relative agreement interaction model

Abstract: We model opinion dynamics in populations of agents with continuous opinion and uncertainty. The opinions and uncertainties are modified by random pair interactions. We propose a new model of interactions, called relative agreement model, which is a variant of the previously discussed bounded confidence. In this model, uncertainty as well as opinion can be modified by interactions. We introduce extremist agents by attributing a much lower uncertainty (and thus higher persuasion) to a small proportion of agents at the extremes of the opinion distribution. We study the evolution of the opinion distribution submitted to the relative agreement model. Depending upon the choice of parameters, the extremists can have a very local influence or attract the whole population. We propose a qualitative analysis of the convergence process based on a local field notion. The genericity of the observed results is tested on several variants of the bounded confidence model.

Some approximation theorems via statistical convergence

Abstract: In this paper we prove some Korovkin and Weierstrass type approximation theorems via statistical convergence. We are also concerned with the order of statistical convergence of a sequence of positive linear operators.

GO‐GARCH: a multivariate generalized orthogonal GARCH model

Abstract: Multivariate GARCH specifications are typically determined by means of practical considerations such as the ease of estimation, which often results in a serious loss of generality. A new type of multivariate GARCH model is proposed, in which potentially large covariance matrices can be parameterized with a fairly large degree of freedom while estimation of the parameters remains feasible. The model can be seen as a natural generalization of the O-GARCH model, while it is nested in the more general BEKK model. In order to avoid convergence difficulties of estimation algorithms, we propose to exploit unconditional information first, so that the number of parameters that need to be estimated by means of conditional information is more than halved. Both artificial and empirical examples are included to illustrate the model. Copyright © 2002 John Wiley & Sons, Ltd.

On the Global Convergence of a Filter--SQP Algorithm

Abstract: A mechanism for proving global convergence in SQP--filter methods for nonlinear programming (NLP) is described. Such methods are characterized by their use of the dominance concept of multiobjective optimization, instead of a penalty parameter whose adjustment can be problematic. The main point of interest is to demonstrate how convergence for NLP can be induced without forcing sufficient descent in a penalty-type merit function. The proof relates to a prototypical algorithm, within which is allowed a range of specific algorithm choices associated with the Hessian matrix representation, updating the trust region radius, and feasibility restoration.

Convergence of Adaptive Finite Element Methods

Abstract: Adaptive finite element methods (FEMs) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

A projection neural network and its application to constrained optimization problems

Abstract: In this paper, we present a recurrent neural network for solving the nonlinear projection formulation. It is shown here that the proposed neural network is stable in the sense of Lyapunov and globally convergent, globally asymptotically stable, and globally exponentially stable, respectively under different conditions. Compared with the existing neural network for solving the projection formulation, the proposed neural network has a single-layer structure and is amenable to parallel implementation. Moreover, the proposed neural network has no Lipschitz condition, and, thus can be applied to solve a very broad class of constrained optimization problems that are special cases of the nonlinear projection formulation. Simulation shows that the proposed neural network is effective in solving these constrained optimization problems.

Time and frequency domain convergence properties in iterative learning control

Abstract: The convergence properties of iterative learning control (ILC) algorithms are considered. The analysis is carried out in a framework using linear iterative systems, which enables several results from the theory of linear systems to be applied. This makes it possible to analyse both first-order and high-order ILC algorithms in both the time and frequency domains. The time and frequency domain results can also be tied together in a clear way. Results are also given for the iterationvariant case, i.e. when the dynamics of the system to be controlled or the ILC algorithm itself changes from iteration to iteration.

A continuous-time observer which converges in finite time

Abstract: It is shown that a continuous-time observer, which comprises two standard nth order observers and a delay D, can observe the state of an nth order linear system in finite time D exactly. In particular, (almost) any convergence time D can be assigned, independent of the observer eigenvalues.

Route flap damping exacerbates internet routing convergence

Abstract: Route flap damping is considered to be a widely deployed mechanism in core routers that limits the widespread propagation of unstable BGP routing information. Originally designed to suppress route changes caused by link flaps, flap damping attempts to distinguish persistently unstable routes from routes that occasionally fail. It is considered to be a major contributor to the stability of the Internet routing system.We show in this paper that, surprisingly, route flap damping can significantly exacerbate the convergence times of relatively stable routes. For example, a route to a prefix that is withdrawn exactly once and re-announced can be suppressed for up to an hour (using the current RIPE recommended damping parameters). We show that such abnormal behavior fundamentally arises from the interaction of flap damping with BGP path exploration during route withdrawal. We study this interaction using a simple analytical model and understand the impact of various BGP parameters on its occurrence using simulations. Finally, we outline a preliminary proposal to modify route flap damping scheme that removes the undesired interaction in all the topologies we studied. .

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Abstract: We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.

On choosing and bounding probability metrics

Abstract: When studying convergence of measures, an important issue is the choice of probability metric. In this review, we provide a summary and some new results concerning bounds among ten important probability metrics/distances that are used by statisticians and probabilists. We focus on these metrics because they are either well-known, commonly used, or admit practical bounding techniques. We summarize these relationships in a handy reference diagram, and also give examples to show how rates of convergence can depend on the metric chosen.

An extended nonlinear primal-dual interior-point algorithm for reactive-power optimization of large-scale power systems with discrete control variables

Abstract: This paper presents a new algorithm for reactive-power optimization of large-scale power systems involving both discrete and continuous variables. This algorithm realizes successive discretization of the discrete control variables in the optimization process by incorporating a penalty function into the nonlinear primal-dual interior-point algorithm. The principle of handling these discrete variables by the penalty function, the timing of introducing the penalty function during iterations, and the setting of penalty factors are discussed in detail. To solve the high-dimension linear correction equation speedily and efficiently in each iteration, a novel data structure rearrangement is proposed. Compared with the existing data structures, it can effectively reduce the number of nonzero fill-in elements and does not give rise to difficulty in triangular factorization. The numerical results of test systems that range in size from 14 to 538 buses have shown that the proposed method can give nearly optimum solutions, has good convergence, and is suitable for large-scale system applications.

Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients

Abstract: A new active-set method for smooth box-constrained minimization is introduced. The algorithm combines an unconstrained method, including a new line-search which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm.

A primal-dual interior point method for optimal power flow dispatching

Abstract: In this paper, the solution of the optimal power flow dispatching (OPFD) problem by a primal-dual interior point method is considered. Several primal-dual methods for optimal power flow (OPF) have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. The aim of the present work is to enhance convergence through two modifications: a filter technique to guide the choice of the step length and an altered search direction in order to avoid convergence to a nonminimizing stationary point. A reduction in computational time is also gained through solving a positive definite matrix for the search direction. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.

Improving BGP convergence through consistency assertions

Abstract: This paper presents a new mechanism for improving the convergence properties of path vector routing algorithms, such as BGP. Using a route's path information, we develop two consistency assertions for path vector routing algorithms that are used to compare similar routes and identify infeasible routes. To apply these assertions in BGP, mechanisms to signal failure/policy withdrawal, and traffic engineering are provided. Our approach was implemented and deployed in a BGP testbed and evaluated using simulation. By identifying and ignoring the infeasible routes, we achieved substantial reduction in both BGP convergence time and the total number of intermediate route changes.

Observers for Lipschitz non-linear systems

Abstract: In an interesting paper R. Rajamani and Y. M. Cho have proposed a systematic methodology to design observers. They have introduced a new problem: relation between distance to unobservability and convergence of 'Luenberger-like' observers. A result for the convergence of the observer has also been given. They have presented a quantity denoted by i, claimed as the distance to unobservability. We show that this number is not actually the distance from the system to the set of unobservable systems. Moreover, the result for the observer is incorrect. We provide a counterexample for the result of convergence. In this paper results for convergence are obtained, with additional strengthened hypothesis, to correct Rajamani and Cho's result. The results are used to design an observer for a, now classical, single-link flexible robot joint. The behaviour of the observer to noisy output is quite satisfactory.

A structural EM algorithm for phylogenetic inference.

Abstract: A central task in the study of molecular evolution is the reconstruction of a phylogenetic tree from sequences of current-day taxa. The most established approach to tree reconstruction is maximum likelihood (ML) analysis. Unfortunately, searching for the maximum likelihood phylogenetic tree is computationally prohibitive for large data sets. In this paper, we describe a new algorithm that uses Structural Expectation Maximization (EM) for learning maximum likelihood phylogenetic trees. This algorithm is similar to the standard EM method for edge-length estimation, except that during iterations of the Structural EM algorithm the topology is improved as well as the edge length. Our algorithm performs iterations of two steps. In the E-step, we use the current tree topology and edge lengths to compute expected sufficient statistics, which summarize the data. In the M-Step, we search for a topology that maximizes the likelihood with respect to these expected sufficient statistics. We show that searching for better topologies inside the M-step can be done efficiently, as opposed to standard methods for topology search. We prove that each iteration of this procedure increases the likelihood of the topology, and thus the procedure must converge. This convergence point, however, can be a suboptimal one. To escape from such "local optima," we further enhance our basic EM procedure by incorporating moves in the flavor of simulated annealing. We evaluate these new algorithms on both synthetic and real sequence data and show that for protein sequences even our basic algorithm finds more plausible trees than existing methods for searching maximum likelihood phylogenies. Furthermore, our algorithms are dramatically faster than such methods, enabling, for the first time, phylogenetic analysis of large protein data sets in the maximum likelihood framework.

Convergence of Iterative Split-Operator Approaches for Approximating Nonlinear Reactive Transport Problems

Abstract: Numerical solutions to nonlinear reactive solute transport problems are often computed using split-operator (SO) approaches, which separate the transport and reaction processes. This uncoupling introduces an additional source of numerical error, known as the splitting error. The iterative split-operator (ISO) algorithm removes the splitting error through iteration. Although the ISO algorithm is often used, there has been very little analysis of its convergence behavior. This work uses theoretical analysis and numerical experiments to investigate the convergence rate of the iterative split-operator approach for solving nonlinear reactive transport problems.

A Numerical Scheme for Impact Problems I: The One-Dimensional Case

Abstract: We consider a mechanical system with impact and one degree of freedom. The system is not necessarily Lagrangian. The representative point is subject to the constraint $u(t) \in \Er^+$ for all t. We assume that, at impact, the velocity is reversed and multiplied by a given coefficient of restitution $e\in[0,1]$. We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is an ad hoc scheme which does not require a systematic search for the times of impact. We prove the convergence of this numerical scheme to a solution. Many of the features of this proof will be reused in the nonconvex, multidimensional case, written in generalized coordinates, given in the companion paper [L. Paoli and M. Schatzman, SIAM J. Numer. Anal., 40 (2002), pp. 734--768]. We present some numerical results obtained with the scheme for a spring-dashpot system and we compare them to the results obtained by impact detection and penalization.

On the Global Convergence of Derivative-Free Methods for Unconstrained Optimization

Abstract: In this paper, starting from the study of the common elements that some globally convergent direct search methods share, a general convergence theory is established for unconstrained minimization methods employing only function values. The introduced convergence conditions are useful for developing and analyzing new derivative-free algorithms with guaranteed global convergence. As examples, we describe three new algorithms which combine pattern and line search approaches.

A new state observer for perspective systems

Abstract: In this note, we consider the problem of estimating the state of a class of perspective systems. The problem can be converted into the observation of a dynamical system with nonlinearities. A new discontinuous state observer, which is motivated by the sliding mode control method and adaptive techniques, is proposed for the obtained dynamical system. The assumptions are reasonable, and the convergence conditions are intuitive and have apparently physical interpretations. The attraction of the new method is that the algorithm is very simple and easy to be implemented, and it is robust to measurement noises. Further, minor a priori knowledge of the system is required in the new formulation. Simulation results show the superiority of the new method to the traditional ones.