Conditions For Global Convergence Research Articles

Distributed Rate Allocation for Inelastic Flows

A common assumption behind most of the recent research on network rate allocation is that traffic flows are elastic, which means that their utility functions are concave and continuous and that there is no hard limit on the rate allocated to each flow. These critical assumptions lead to the tractability of the analytic models for rate allocation based on network utility maximization, but also limit the applicability of the resulting rate allocation protocols. This paper focuses on inelastic flows and removes these restrictive and often invalid assumptions. First, we consider nonconcave utility functions, which turn utility maximization into difficult, nonconvex optimization problems. We present conditions under which the standard price-based distributed algorithm can still converge to the globally optimal rate allocation despite nonconcavity of utility functions. In particular, continuity of price-based rate allocation at all the optimal prices is a sufficient condition for global convergence of rate allocation by the standard algorithm, and continuity at at least one optimal price is a necessary condition. We then show how to provision link capacity to guarantee convergence of the standard distributed algorithm. Second, we model real-time flow utilities as discontinuous functions. We show how link capacity can be provisioned to allow admission of all real-time flows, then propose a price-based admission control heuristics when such link capacity provisioning is impossible, and finally develop an optimal distributed algorithm to allocate rates between elastic and real-time flows.

Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets

A condition for global convergence of a homotopy method for a variational inequality problem (VIP) on an unbounded set is introduced. The condition is derived from a concept of a solution at infinity to VIP. By an argument of the existence of a homotopy path, we show that VIP has a solution if it has no solution at infinity. It is proved that if any of several well-known conditions given in the literature holds, there is no solution at infinity. Furthermore, a globally convergent homotopy method is developed to compute a solution to VIP. Several numerical examples illustrate how to follow the homotopy path starting at an arbitrary point in the unbounded set.

Comments on “A Blind Signal Separation Method for Multiuser Communications”

This correspondence first shows that the global convergence analysis of the method proposed in the above paper is incomplete. Then we provide a counter example to show that the sufficient condition for global convergence is incorrect

Parameter-Dependent Lyapunov Function Method for a Class of Uncertain Nonlinear Systems with Multiple Equilibria

First, a new linear matrix inequality (LMI) characterization of extended strict positive realness is presented for linear continuous-time systems. Then a class of nonlinear systems with multiple equilibria subject to polytopic uncertainty is addressed by the parameter-dependent Lyapunov function method. New sufficient conditions for global convergence are presented. This allows the Lyapunov function to be parameter dependent. Furthermore, an LMI-based controller design method is also given, and reduced-order controllers can be designed by performing a structural constraint on the introduced slack variables. Several numerical examples are included to demonstrate the applicability of the proposed method.

Convergence of the Primal‐Dual Active Set Strategy for Diagonally Dominant Systems

Sufficient conditions for global convergence of the primal-dual active set strategy for finite and infinite dimensional quadratic, as well as nonlinear optimization, problems with affine equality and inequality constraints are presented. These conditions involve diagonal dominance and cone preserving properties of the operator defining the cost functional. Globalization strategies are also provided, and specific sufficient conditions for the primal-dual active set step to have a descent property are given.

Intergenerational mobility and macroeconomic history dependence

A large literature on ‘endogenous inequality’ has argued that persistent differences in macroeconomic performance across countries can be explained by historical inequality, owing to indivisibilities in occupational choice and borrowing constraints. These models are characterized by homogenous agents, a continuum of steady states (SSs) and lack of mobility in every SS. We show that introducing (even a little) heterogeneity in order to generate SS mobility shrinks the SS set dramatically. Mobile SSs are generically locally unique and finite in number. Sufficient conditions for global uniqueness and convergence of competitive equilibrium dynamics are provided.

Evolutionarily stable strategies and short-term selection in Mendelian populations re-visited

This note concerns a one locus, two allele, random mating diploid population, subject to frequency-dependent viability selection. It is already known that in such a population, any evolutionarily stable strategies (ESS), if only accessible by the genotype-to-phenotype mapping, is the phenotypic image of a stable genetic equilibrium (Eshel, I. 1982. Evolutionarily stable strategies and viability selection in Mendelian populations. Theor. Popul. Biol. 22(2), 204–217; Cressman et al. 1996. Evolutionary stability in strategic models of single-locus frequency-dependent viability selection. J. Math. Biol. 34, 707–733). The opposite is not true. We find necessary and sufficient parametric conditions for global convergence to the ESS, but we also demonstrate conditions under which, although a unique, genetically accessible ESS exists, there is another, “non-phenotypic” genetically stable equilibrium.

Global exponential convergence of multitime-scale neural networks

In this paper, we investigate the convergence and stability of a neural network model with different time scales, which models the activity of cortical cognitive maps. We provide a theoretic condition for global exponential convergence of the solutions of the network, which is proved weaker than some existing results in the literature. We also introduce time-varying delays with less constraints into the neural network model and derive a general stability condition for the delay network.

On the global output convergence of a class of recurrent neural networks with time-varying inputs

This paper studies the global output convergence of a class of recurrent neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions and locally Lipschitz continuous time-varying inputs. We establish two sufficient conditions for global output convergence of this class of neural networks. Symmetry in the connection weight matrix is not required in the present results which extend the existing ones.

The Possibility of Cyclical Behavior in a Class of Dynamic Models

The study investigates conditions under which endo genous cyclic behavior may be observed within the context of Predator-Prey (Lotka -Volterra) Models. The analysis also establishes conditions under which such behavior is non-existen t and hence establishes conditions for global convergence to the interior equilibrium, whenever i t exists. The results are then applied to two diverse sets of economic exercises and shows how th e conclusions of those exercises may be established under a much weaker set of assumptions. Based on these discussions, a numerical example of robust periodic behavior is provided.

Multipoint cubic surrogate function for sequential approximate optimization

Multipoint cubic approximations are investigated as surrogate functions for nonlinear objective and constraint functions in the context of sequential approximate optimization. The proposed surrogate functions match actual function and gradient values, including the current expansion point, thus satisfying the zero and first-order necessary conditions for global convergence to a local minimum of the original problem. Function and gradient information accumulated from multiple design points during the iteration history is used in estimating a reduced Hessian matrix and selected cubic terms in a design subspace appropriate for problems with many design variables. The resulting approximate response surface promises to accelerate convergence to an optimal design within the framework of a trust region algorithm. The hope is to realize computational savings in solving large numerical optimization problems. Numerical examples demonstrate the effectiveness of the new multipoint surrogate function in reducing errors over large changes in design variables.

Increasing the Robustness of a Preconditioned Filtered-X LMS Algorithm

This letter presents a robustification of the preconditioned filtered-X LMS algorithm proposed by Elliott et al. The method optimizes the average performance for probabilistic uncertainty in the secondary path and relaxes the SPR condition for global convergence. It also prevents large amplification in the preconditioning filters due to secondary path zeros on and/or close to the unit circle, which may yield overactuation in practical applications.

A penalty method for American options with jump diffusion processes

The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence.

Convergence analysis of the Filtered-U LMS algorithm for active noise control in case perfect cancellation is not possible

The Filtered-U LMS algorithm, proposed by Eriksson for active noise control applications, adapts the coefficients of an infinite-impulse response controller. Conditions for global convergence of the Filtered-U LMS algorithm were presented by Wang and Ren (Signal Processing, 73 (1999) 3) and Mosquera and Pérez-González (Signal Processing, 80 (2000) 5) for the case where perfect noise cancellation is achievable, which means only measurement noise remains. This paper shows that the assumption of perfect cancellation is not necessary. In real situations perfect cancellation is often not achievable due to delays and non-minimum phase zeros. The conclusion is derived by analysis of the structure of the Wiener optimal solution. This also leads to the suggestion of preconditioning filters in the Filtered-U LMS updating. The preconditioning has shown considerable increase of the convergence rate in a realistic simulation study.

Local tuning and partition strategies for diagonal GO methods

In this paper, global optimization (GO) Lipschitz problems are considered where the multi-dimensional multiextremal objective function is determined over a hyperinterval. An efficient one-dimensional GO method using local tuning on the behavior of the objective function is generalized to the multi-dimensional case by the diagonal approach using two partition strategies. Global convergence conditions are established for the obtained diagonal geometric methods. Results of a wide numerical comparison show a strong acceleration reached by the new methods working with estimates of the local Lipschitz constants over different subregions of the search domain in comparison with the traditional approach.

An efficient algorithm for the zero crossing detection in digitized measurement signal

A new algorithm for fast determining the first zero crossing point over an interval is proposed. This problem arises in a number of applications, for example, in the determination of the phase difference between two signals. The method uses only the values of the objective function in its work. The algorithm works also in the cases where there is no any zero crossing point over the search interval, and determines the global minimum of the objective function. Global convergence conditions are established for the new algorithm. Numerical experiments and comparison with methods taken from literature show that the new technique is fast, reliable, and stable to high levels of noise. Finally, experimental results coming out from applications confirm the validity of the proposed algorithm.

Comments on "Closed-form analysis of linearly constrained CMA-based blind multiuser detector"

We point out that the proof of global convergence of LCCM detector is incorrect in the paper by Tang et al. (see ibid., vol.4, no.9, p.273-76, 2000). According to propositions presented in the paper by Xu and Feng (see Electron. Lett., vol.36, no.2, p.171-72, 2000), we further analyze the conditions for global convergence of LCCM detector, and give the low bound of constraint on the power of desired user.

Convergence analysis of cellular neural networks with unbounded delay

Cellular neural networks (CNNs) have been successfully applied in many areas such as classification of patterns, image processing, associative memories, etc. Since they are inherently local in nature, they can be easily implemented in very large scale integration. In the processing of static images, CNNs without delay are often applied whereas in the processing of moving images, CNNs with delay have been found more suitable. This paper proposes a more general model of CNNs with unbounded delay, which may have potential applications in processing such motion related phenomena as moving images, and studies global convergence properties of this model. The dynamic behaviors of CNNs, especially their convergence properties, play important roles in applications. This paper: (1) introduces a class of CNNs with unbounded delay; (2) gives some interesting properties of a network's output function; (3) establishes relationships between a network's state stability and its output stability; and (4) obtains simple and easily checkable conditions for global convergence by functional differential equation methods.

New theorems on global convergence of some dynamical systems

In this paper, we discuss dynamical behaviors of recurrently asymmetrically connected neural networks. We propose a new approach to study global convergence of the networks. Better test conditions for global convergence are given.

Parallel Asynchronous Global Search and the Nested Optimization Scheme

In this paper, a parallel asynchronous information algorithm for solving multidimensional Lipschitz global optimization problems, where times for evaluating the objective function can be different from point to point, is proposed. This method uses the nested optimization scheme and a new parallel asynchronous global optimization method for solving core univariate subproblems generated by the nested scheme. The properties of the scheme related to parallel computations are investigated. Global convergence conditions for the new method and theoretical conditions of speed up, which can be reached by using asynchronous parallelization in comparison with the pure sequential case, are established. Numerical experiments comparing sequential, synchronous, and asynchronous algorithms are also reported.