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We implement and test a globally convergent sequential approximate optimization algorithm based on (convexified) diagonal quadratic approximations. The algorithm resides in the class of globally convergent optimization methods based on conservative convex separable approximations developed by Svanberg. At the start of each outer iteration, the initial curvatures of the diagonal quadratic approximations are estimated using historic objective and/or constraint function value information, or by building the diagonal quadratic approximation to the reciprocal approximation at the current iterate. During inner iterations, these curvatures are increased if no feasible descent step can be made. Although this conditional enforcement of conservatism on the subproblems is a relaxation of the strict conservatism enforced by Svanberg, global convergence is still inherited from the conservative convex separable approximations framework developed by Svanberg. A numerical comparison with the globally convergent version of the method of moving asymptotes and the nonconservative variants of both our algorithm and method of moving asymptotes is made. DOI: 10.2514/1.41975

The increasing complexity of engineering systems has motivated continuing research on computational learning methods toward making autonomous intelligent systems that can learn how to improve their performance over time while interacting with their environment. These systems need not only to sense their environment, but also to integrate information from the environment into all decision-makings. The evolution of such systems is modeled as an unknown controlled Markov chain. In a previous research, the predictive optimal decision-making (POD) model was developed, aiming to learn in real time the unknown transition probabilities and associated costs over a varying finite time horizon. In this paper, the convergence of the POD to the stationary distribution of a Markov chain is proven, thus establishing the POD as a robust model for making autonomous intelligent systems. This paper provides the conditions that the POD can be valid, and be an interpretation of its underlying structure. [DOI: 10.1115/1.3117202]

In this technical note, we propose a new fixed-lag smoother that estimates the fixed-delayed state for a stochastic continuous-time system. The estimation error variance of the proposed smoother is minimized under the constraint that the estimated state converges to the real state exactly in finite time after noises or uncertainties disappear. For numerical computing, the proposed smoother is represented in a recursive form. Unlike other approaches, any additional processes such as batch processing and sampling data through discrete-time techniques are not required to achieve the finite time convergence. A numerical example is presented to illustrate the finite time convergence of the proposed smoother in comparison with the asymptotic convergence of an optimal smoother. Index Terms--Asymptotic convergence, finite time convergence, fixed-lag smoother.

This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples. Index Terms--Convergence, neural network, stability, variational inequality.

This paper studies the weights stability and accuracy of the implicit fifth-order weighted essentially nonoscillatory finite difference scheme. It is observed that the weights of the Jiang-Shu weighted essentially nonoscillatory scheme oscillate even for smooth flows. An increased [epsilon] value of [10.sup.-2] is suggested for the weighted essentially nonoscillatory smoothness factors, which removes the weights oscillation and significantly improves the accuracy of the weights and solution convergence. With the improved [epsilon] value, the weights achieve the optimum value with minimum numerical dissipation in smooth regions and maintain the sensitivity to capture nonoscillatory shock profiles for the transonic flows. The theoretical justification of this treatment is given in the paper. The wall surface boundary condition uses a half-point mesh so that the conservative differencing can be enforced. A third-order accurate finite difference scheme is given to treat wall boundary conditions. The implicit time-marching method with unfactored Gauss--Seidel line relaxation is used with the high-order schemes to achieve a high convergence rate. Several transonic cases are calculated to demonstrate the robustness, efficiency, and accuracy of the methodology.

Recent research has demonstrated the success of supervised dimensionality reduction algorithms 2DLDA and 2DMFA, which are based on the image-as-matrix representation, in small sample size cases. To solve the convergence problem in 2DLDA and 2DMFA, we propose in this work two new schemes, called Null Space based 2DLDA (NS2DLDA) and Null Space based 2DMFA (NS2DMFA), and apply them to the challenging multi-view face recognition task. First, we convert each 2-D face image (matrix) into a vector and compute the first projection matrix P1 from the null space of the intra-class scatter matrix, such that the samples from the same class are projected to the same point. Then the data are projected and reconstructed with P1. Finally, we re-organize the reconstructed datum into a matrix and then compute the second projection direction P2, in the form of a Kronecker product of two matrices, by maximizing the inter-class scatter. A proof of algorithmic convergence is provided. The experiments on two benchmark multi-view face databases, the CMU PIE and FERET databases, demonstrate that NS2DLDA outperforms Fisherface, Null Space LDA (NSLDA) and 2DLDA. Additionally, NS2DMFA is also demonstrated to be more accurate than MFA and 2DMFA for face recognition. Index Terms--LDA, MFA, multiview face recognition, null space LDA, 2DLDA, 2DMFA.

A controller design methodology for linear continuous-time systems using a state observer is proposed. In particular, the objective is to smooth the discontinuity resulting from a piecewise linear control law previously proposed, which improves the convergence rate of systems under asymmetrically constrained control. The proposed method is based on the use of a non quadratic asymmetrical Lyapunov function. Both full-order observers and minimal order observers are considered to reconstruct the inaccessible state. The speed of convergence of the controlled system and the observer is improved, with smooth control vector components, that respect always the imposed asymmetric constraints. Index Terms--Asymmetric constrained control, convergence rate improvement, decay rate, linear systems, state observers.

In this paper, a recurrent neutral network for both convex and nonconvex equality-constrained optimization problems is proposed, which makes use of a cost gradient projection onto the tangent space of the constraints. The proposed neural network constructs a generically nonfeasible trajectory, satisfying the constraints only as t [right arrow] [infinity]. Local convergence results are given that do not assume convexity of the optimization problem to be solved. Global convergence results are established for convex optimization problems. An exponential convergence rate is shown to hold both for the convex case and the nonconvex case. Numerical results indicate that the proposed method is efficient and accurate. Index Terms--Constrained optimization, convergence, convex and nonconvex problems, recurrent neural networks.

A formal theoretical explanation of the model-mismatch instability problem associated with certain adaptive control design schemes is proposed, and a solution is provided. To address the model-mismatch problem, a primary task of adaptive control is formulated as finding an asymptotically optimal, stabilizing controller, given the feasibility of adaptive control problem. A dass of data-driven cost functions called cost-detectable is intro. dnced that detect evidence of instability without reference to prior plant models or plant assumptions. The problem of designing adaptive systems that are robustly immune to mismatch instability problems is thus placed in a setting of a standard optimization problem. We call the result safe adaptive control because it robustly achieves adaptive stabilization goals whenever feasible, without prior assumptions on the plant model and, hence, without the risk of model-mismatch instability. The result improves the robustness of previous results in hysteresis switching control, both for discrete and for continuously-parameterized candidate controller sets. Examples are provided. Index Terms--Adaptive control, convergence, robustness, stability, switching.

This brief paper considers iterative learning control (ILC) for precision motion control (PMC) applications. This work develops a methodology to design a low pass filter, called the Q-filter, that is used to limit the bandwidth of the ILC to prevent the propagation of high frequencies in the learning. A time-varying bandwidth Q-filter is considered because PMC reference trajectories can exhibit rapid changes in acceleration that may require high bandwidth for short periods of time. Time-frequency analysis of the initial error signal is used to generate a shape ,function for the bandwidth profile. Key parameters of the bandwidth profile are numerically optimized to obtain the best tradeoff in converged error and convergence speed. Simulation and experimental results for a permanent-magnet linear motor are included. Results show that the optimal time-varying Q-filter bandwidth provides faster convergence to lower error than the optimal time-invariant bandwidth. [DOI: 10.1115/1.2936844]

The problem of estimating unknown frequencies of a sinusoidal signal simultaneously is a classical problem in signal and system theory. Many approaches and algorithms are proposed in literature to develop estimators for a measurable sinusoidal signal having multiple sinusoids with unknown amplitudes, frequencies and phases. In this work, an asymptotically convergent frequency estimator is given for estimation of n-unknown frequencies of a measurable sinusoidal signal. The contraction theory approach is adopted to show the asymptotic convergence of the proposed estimator in quite a simplified manner. Approach given here exploits the results of contraction theory related to semi-contracting systems. The nonlinear estimator based on dynamic system approach, guarantees global boundedness and convergence of the state and frequency estimation for all initial conditions and frequency values. It further ensures simultaneous globally convergent estimation of states and frequencies of a sinusoid involving multiple frequencies. Numerical simulations are presented for different combinations of frequencies to justify the claim. Index Terms--Contraction theory, frequency estimator, exponential convergence, semi-contracting, sinusoidal signal.

We introduce a framework for fault-tolerant supervisory control of discrete-event systems. Given a plant, possessing both faulty and nonfaulty behavior, and a submodel for just the nonfaulty part, the goal of fault-tolerant supervisory control is to enforce a certain specification for the nonfaulty plant and another (perhaps more liberal) specification for the overall plant, and further to ensure that the plant recovers from any fault within a bounded delay so that following the recovery the system state is equivalent to a nonfaulty state (as if no fault ever happened). The specification for the overall plant is more liberal compared to the one for the nonfaulty part since a degraded performance may be allowed after a fault has occurred. We formulate this notion of fault-tolerant supervisory control and provide a necessary and sufficient condition for the existence of such a supervisor. The condition involves the usual notions of controllability, observability and relative-closure, together with the notion of stability. An example of a power system is provided to illustrate the framework. We also propose a weaker notion of fault-tolerance where following the recovery, the system state is simulated by some nonfaulty state, i.e., behaviors following the recovery are also the behaviors from some faulty state. Also, we formulate the corresponding notion of weakly fault-tolerant supervisory control and present a necessary and sufficient condition (involving the notion of language-stability) for the its existence. We also introduce the notion of nonuniformly-bounded fault-tolerance (and its weak version) where the delay-bound for recovery is not uniformly bounded over the set of faulty traces, and show that when the plant model has finitely many states, this more general notion of fault-tolerance coincides with the one in which the delay-bound for recovery is uniformly bounded. Index Terms--Discrete event systems, fault-tolerance, language convergence, stability, supervisory control.

The economic dispatch has the objective of generation allocation to the power generators in such a manner that the total fuel cost is minimized while all operating constraints are satisfied. Conventional optimization methods assume generator cost curves to be continuous and monotonically increasing, but modern generators have a variety of nonlinearities in their cost curves making this assumption inaccurate, and the resulting approximate dispatches cause a lot of revenue loss. Evolutionary methods like particle swarm optimization perform better for such problems as no convexity assumptions are imposed, but these methods converge to sub-optimum solutions prematurely, particularly for multimodal problems. To handle the problem of premature convergence, this paper proposes to apply a novel self-organizing hierarchical particle swarm optimization (SOH_PSO) for the nonconvex economic dispatch (NCED). The performance further improves when time-varying acceleration coefficients are included. The results show that the proposed approach outperforms previous methods for NCED. Index Terms--Nonconvex economic dispatch (NCED), premature convergence, prohibited operating zones (POZ), ramp rate limits, self-organizing hierarchical particle swarm optimization (SOH_PSO), time-varying acceleration coefficients (TVAC), valve point loading effect.

In this correspondence, we show that a function in the Bernstein class [B.sup.p.sub.[pi]] (1 [less than or equal to] p [less than or equal to] [infinity]) can be reconstructed in the Lagrange interpolation form with sampling points {[t.sub.n]} satisfying [absolute value of [t.sub.n] - n] [less than or equal to] d and [4d] additional points, where d is any nonnegative real number that is not necessarily less than 1/4. Here, [x] denotes the integer part of x. Index Terms--Bandlimited function, Kadec's constant 1/4, Lagrange interpolation, nonuniform sampling series expansion, uniform convergence.

In this paper, we propose a distributed parallel support vector machine (DPSVM) training mechanism in a configurable network environment for distributed data mining. The basic idea is to exchange support vectors among a strongly connected network (SCN) so that multiple servers may work concurrently on distributed data set with limited communication cost and fast training speed. The percentage of servers that can work in parallel and the communication overhead may be adjusted through network configuration. The proposed algorithm further speeds up through online implementation and synchronization. We prove that the global optimal, classifier can be achieved iteratively over an SCN. Experiments on a real-world data set show that the computing time scales well with the size of the training data for most networks. Numerical results show that a randomly generated SCN may achieve better performance than the state of the art method, cascade SVM, in terms of total training time. Index Terms--Convergence, distributed data mining, parallel computing, strongly connected networks (SCNs), support vector machine (SVM).

Thispaper presents a biconjugate gradient (BiCG) method that can significantly improve the performance of the quasi-static finite-difference scheme, which has been widely used to model field induction phenomena in voxel phantoms. The proposed BiCG method offers remarkable computational advantages in terms of convergence performance and memory consumption over the conventional iterative, successive overrelaxation algorithm. The scheme has been validated against other known solutions on a lossy, multilayered ellipsoid phantom excited by an ideal coil loop. The wide application capability and computational performance of the BiCG method is demonstrated by modeling the exposure of MRI healthcare workers to fields produced by pulsed field gradients. This is an important topic of research in light of the Physical Agents Directive 2004/40/EC because a variety of realistic operator postures near the bore entrance of an MRI system are modeled. Index Terms--Biconjugate gradient (BiCG) method, MRI, quasi-static finite-difference (QSFD), successive overrelaxation (SOR).

Although the convergence behavior of gradient-based adaptive algorithms, such as steepest descent and leas mean square (LMS), has been extensively studied, the influence of the desired response on the transient convergence has generally received little attention. However, empirical results show that this signal can have a great impact on the learning curve. In this paper we analyze the influence of the desired response on the transient convergence by making a novel interpretation, from the viewpoint of the desired response, of previous convergence analyses of SD and LMS algorithms. We show that, without prior knowledge that can be used to wisely select the initial weight vector, initial convergence is fast whenever there is high similarity between input and desired response whereas, on the contrary, when there is low similarity between these two signals, convergence is slow from the beginning. Index Terms--Adaptive filters, adaptive signal processing, convergence, gradient methods, least-mean-square (LMS) methods.

The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the well-known sign-flipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the so-called QR FastICA algorithm, which employs the QR decomposition (Gram-Schmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA. Index Terms--Approximate Newton method, convergence analysis, FastICA, independent component analysis (ICA), principal fiber bundle.

Global convergence of the sequential minimal optimization (SMO) algorithm for support vector regression (SVR) is studied in this paper. Given I training samples, SVR is formulated as a convex quadratic programming (QP) problem with I pairs of variables. We prove that if two pairs of variables violating the optimality condition are chosen for update in each step and subproblems are solved in a certain way, then the SMO algorithm always stops within a finite number of iterations after finding an optimal solution. Also, efficient implementation techniques for the SMO algorithm are presented and compared experimentally with other SMO algorithms. Index Terms--Convergence, quadratic programming (QP), sequential minimal optimization (SMO), support vector regression (SVR).

A fast blind adaptive beamforming algorithm is proposed in code-division multiple-access wireless systems based on an array pattern synthesis approach. By incorporating an interference suppression mechanism in the iterative procedure, the proposed algorithm outperforms conventional blind algorithms. Two practical adaptive beamforming algorithms are further presented, which provide fast convergence speed with low complexity. The simulation results demonstrate the effectiveness of the proposed adaptive beamforming algorithms in various signal environments. Index Terms--Adaptive beamforming, blind algorithm, fast convergence, low complexity.

The cerebellum constitutes a vital part of the human brain system that possesses the capability to model highly nonlinear physical dynamics. The cerebellar model articulation controller (CMAC) associative memory network is a computational model inspired by the neuropbysiological properties of the cerebellum, and it has been widely used for control, optimization, and various pattern recognition tasks. However, the CMAC network's highly regularized computing structure often leads to the following: l) a suboptimal modeling accuracy, 21 poor memory utilization, and 3) the generalization-accuracy dilemma. Previous attempts to address these shortcomings have limited success and the proposed solutions often introduce a high operational complexity to the CMAC network. This paper presents a novel neurophysiologically inspired associative memory architecture named pseudo-self-evolving CMAC (PSECMAC) that nonuniformly allocates its computing cells to overcome the architectural deficiencies encountered by the CMAC network. The nonuniform memory allocation scheme employed by the proposed PSECMAC network is inspired by the cerebellar experience-driven synaptic plasticity phenomenon observed in the cerebellum, where significantly higher densities of synaptic connections are located in the frequently accessed regions. In the PSECMAC network, this biological synaptic plasticity phenomenon is emulated by employing a data-driven adaptive memory quantization scheme that defines its computing structure. A neighborhood-based activation process is subsequently implemented to facilitate the learning and computation of the PSECMAC structure. The training stability of the PSECMAC network is theoretically assured by the proof of its learning convergence, which will be presented in this paper. The performance of the proposed network is subsequently benchmarked against the CMAC network and several representative CMAC variants on three real-life applications, namely, pricing of currency futures option, banking failure classification, and modeling of the glucose-insulin dynamics of the human glucose metabolic process. The experimental results have strongly demonstrated the effectiveness of the PSECMAC network in addressing the architectural deficiencies of the CMAC network by achieving significant improvements in the memory utilization, output accuracy as well as the generalization capability of the network. Index Terms--Associative memory, brain inspired, cerebellar model articulation controller (CMAC), cerebellum, learning convergence, multiresolution, neural networks, pseudo-self-evolving CMAC (PSECMAC).

In this paper, a one-layer recurrent neural network with a discontinuous hard-limiting activation function is proposed for quadratic programming. This neural network is capable of solving a large class of quadratic programming problems. The state variables of the neural network are proven to be globally stable and the output variables are proven to be convergent to optimal solutions as long as the objective function is strictly convex on a set defined by the equality constraints. In addition, a sequential quadratic programming approach based on the proposed recurrent neural network is developed for general nonlinear programming. Simulation results on numerical examples and support vector machine (SVM) learning show the effectiveness and performance of the neural network. Index Terms--Differential inclusion, global convergence, hard-limiting activation function, Lyapunov stability, nonlinear programming, quadratic programming, recurrent neural network.

It is shown that users of frequency-domain numerical simulators should be warned of the danger of using models with adjacent complementary media since such a configuration of materials can be a cause of a numerical instability which cannot be overcome by the presence of other lossy media or absorbing boundary conditions. The physical realizability at any frequency of models considering complementary media with sharp interfaces is then questioned by the contradiction between the deduced result and the one which holds true when the actual dielectric structure of metamaterials is considered. Index Terms--Boundary value problems, convergence of numerical methods, finite element methods, frequency domain analysis, metamateriais, numerical stability, waveguide discontinuity problems.

This paper is concerned with numerical methods for stochastic differential games of regime-switching diffusions. Numerical methods using Markov chain approximation techniques are developed. A new proof of the existence of a saddle point for the stochastic differential game is provided. This new proof enables us to treat certain systems with nonseparable (in controls) structure. Convergence of the algorithms is derived by means of weak convergence methods. In addition, examples are also provided for demonstration purposes. Index Terms--Controlled regime-switching diffusion, convergence, Markov chain approximation, saddle point, stochastic differential game.

The geometric convergence ratio, the main focus of a discretized scheme for constrained quadratic control problem was examined. In order to allow for the numerical applications of the developed scheme, discretizing the time interval and using Euler's scheme for its differential constraint obtained a finite dimensional approximation. Applying the penalty function method, an unconstrained problem was obtained on function minimization with bilinear form expression. This finally led to the construction of an operator. The Scheme was applied to a sampled problem and it exhibited geometric convergence ratio, [alpha], in the open interval (0, 1) as depicted in column 6 of Table 1. Key words: Quadratic, discretization, bilinear form expression, associated operator, convergence profile and geometric convergence, INTRODUCTION In (1),(2), the scheme establishing the solution of optimal control problems constrained by evolution equation of the delay type with matrix coefficients was presented, without addressing the geometric convergence [...]

This paper is concerned with the global convergence of the solutions of a nonautonomous system with variable delays, arising from the description of the states of neurons in delayed Cohen-Grossberg type in a time-varying situation. By exploring intrinsic features between nonautonomous system and its asymptotic equation, several novel sufficient conditions are established to ensure that all solutions of the networks converge to a periodic function or a constant vector for delayed Cohen-Grossberg-type neural network INN) models in time-varying situation. The results can be applied directly to group of NNs models including Hopfield NNs, bidirectional association memory NNs, and cellular NNs. Our results are not only presented in terms of system parameters and can be easily verified but also are less restrictive than previously known criteria. Numerical simulations have also been presented to demonstrate the theoretical analysis. Index Terms--Convergence, delay, equilibrium, neural networks (NNs), period.

This paper investigates the learning of a wide class of single-hidden-layer feedforward neural networks (SLFNs) with two sets of adjustable parameters, i.e., the nonlinear parameters in the hidden nodes and the linear output weights. The main objective is to both speed up the convergence of second-order learning algorithms such as Levenberg-Marquardt (LM), as well as to improve the network performance. This is achieved here by reducing the dimension of the solution space and by introducing a new Jacobian matrix. Unlike conventional supervised learning methods which optimize these two sets of parameters simultaneously, the linear output weights are first converted into dependent parameters, thereby removing the need for their explicit computation. Consequently, the neural network (NN) learning is performed over a solution space of reduced dimension. A new Jacobian matrix is then proposed for use with the popular second-order learning methods in order to achieve a more accurate approximation of the cost function. The efficacy of the proposed method is shown through an analysis of the computational complexity and by presenting simulation results from four different examples. Index Terms--Approximation, convergence, Jacobian matrix, single-hidden layer feedforward neural network (SLFN), supervised learning, training accuracy.

This paper investigates the convergence properties and consistency of Extended Kalman Filter (EKF) based simultaneous localization and mapping (SLAM) algorithms. Proofs of convergence are provided for the nonlinear two-dimensional SLAM problem with point landmarks observed using a range-and-bearing sensor. It is shown that the robot orientation uncertainty at the instant when landmarks are first observed has a significant effect on the limit and/or the lower bound of the uncertainties of the landmark position estimates. This paper also provides some insights to the inconsistencies of EKF based SLAM that have been recently observed. The fundamental cause of EKF SLAM inconsistency for two basic scenarios are clearly stated and associated theoretical proofs are provided. Index Terms--Convergence, extended information filter, extended Kalman filter, inconsistency, simultaneous localization and mapping (SLAM).

The usefulness of second-order functional derivatives for resolving convergence properties in adaptive finite-element electromagnetics is investigated. Second-order criteria are proposed to assess the convergence of the global solution for arbitrary discretizations of scalar Helmholtz systems. Effective convergence criteria are introduced and evaluated for some practical applications. Index Terms--Adaptive systems, electromagnetic analysis, error analysis, finite-element methods.

State-space models are used for modeling of many physical and economic processes. An asymptotic distribution theory for the state estimate from a Kalman filter in the absence of the usual Gaussian assumption is presented in [1]. They proved the central limit theorem for state estimators when the random terms in the model have arbitrary distribution. In this study, some convergence rates in the central limit theorem are given. These convergence rates are used for the development of a nonparametric test of the validity of the model. Index Terms: Central limit theorem, convergence rate, Kalman filter, model validity tests, nonparametric tests.

Sequence-to-sequence transformations are examined as a means to accelerate the convergence of iterative finite-element methods; in particular, the Shanks transform, the scalar epsilon algorithm (SEA), and the vector epsilon algorithm (VEA) are applied to dynamic relaxation (DR) solutions. One of the features of these algorithms is that they operate directly on the solution iterates and, therefore, do not require modification of the finite-element code itself. The accelerators are applied to a linear spring system and a cantilever elastica. Nonlinear sequence-to-sequence transformations are found to accelerate convergence by a factor of more than 10 and, thus, to largely overcome a major drawback of iterative finite-element methods, namely their slow rate of convergence. The VEA transformation is especially stable, is invariant with respect to coordinate transformations, and is compatible with typical kinematic constraints. Therefore, it is well suited to the acceleration of finite-element solution iterates.

We consider a two-dimensional stochastic approximations scheme of the Robbins-Monro type which naturally arises in the study of steering policies for Markov decision processes. Making use of a decoupling change of variables, we establish its almost sure convergence by ad-hoc arguments that combine standard results on one-dimensional stochastic approximations with a version of the law of large numbers for martingale differences. We use this direct analysis to guide us in selecting the test function which appears in standard convergence results for multidimensional schemes. Furthermore, although a blind application of the ODE method is not possible here due to a lack of regularity properties, the aforementioned change of variables paves the way for an interpretation of the behavior of solutions to the associated limiting ODE.

A Fenchel's duality theorem-based structure performs constrained adaptive filtering in broadband array processing. The adaptation's dual updation scheme involves the use of a rank-one correction and is smaller than standard updation schemes. The updation scheme operates on the Lagrange multiplier vector. Application of the dual approach to power minimization and beam forming problems yield interval sizes for algorithmic convergence.

An adaptive least mean square (LMS) scheme-based Fourier series algorithm calculates probabilities for periodic signals using noise data. The algorithm is valid for data with varying and invariant time parameters. The algorithm's convergence and overall system decoupling occur when the LMS interval decreases to the smallest possible size.

The convergence properties of a substructure synthesis procedure derived from the Rayleigh-Ritz method is evaluated. Substructure synthesis entails the mathematical modeling of different substructures individually and applying compatibility requirements on the substructures to make it as one whole structure. Using the traditional Rayleigh-Ritz method in solving eigenvalue problems through approximations from admissible functions results in poor convergence. The new method overcomes this problem by using mass and stiffness matrices that gives proof of convergence through the inclusion principle.

In irreducible excitatory networks of analog graded-response neurons, the trajectories of most solutions tend to the equilibria. We derive sufficient conditions for such networks to be globally asymptotically stable. When the network possesses several locally stable equilibria, their location in the phase space is discussed and a description of their attraction basin is given. The results hold even when interunit transmission is delayed. Index Terms--Attraction basin, convergence, delay, recurrent network.

The recently proposed low-complexity reduction-by-composition least-mean-square (LMS) algorithm (RCLMS) costs only half multiplications compared to that of the conventional direct-form LMS algorithm (DLMS). This work intends to characterize its properties and conditions for mean and mean-square convergence. Closed-form mean-square error (MSE) as a function of the LMS step-size [micro] and an extra compensation step-size [Alpha] are derived, which are slightly larger than that of the DLMS algorithm. It is shown, when [micro] is small enough and [Alpha] is properly chosen, the RCLMS algorithm has comparable performance to that of the DLMS algorithm. Simple working rules and ranges for [Alpha] and [micro] to make such comparability are provided. For the algorithm to converge, a tight bound for [Alpha] is also derived. The derived properties and conditions are verified by simulations. Index Terms--Adaptive signal processing, convergence, LMS algorithm.

The concept of eigenfunctions for a linear time-invariant system is emphasized in this short paper. It is pointed out that when the concept of eigenfunctions is used, incorrect results may be obtained if the region of convergence of the system function is overlooked. As a consequence, incorrect solutions can be obtained, as was the case for an example found in a widely used textbook. Index Terms--Eigenfunctions, Laplace transforms, region of convergence.