New Sufficient Conditions For Convergence Research Articles

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's solutions after a finite number of steps, but without necessarily strict decrease at every step, in contrast to the classical stochastic Lyapunov theory. As the first application of this new Lyapunov criterion, we look at the product of any random sequence of stochastic matrices, including those with zero diagonal entries, and obtain sufficient conditions to ensure the product almost surely converges to a matrix with identical rows; we also show that the rate of convergence can be exponential under additional conditions. As the second application, we study a distributed network algorithm for solving linear algebraic equations. We relax existing conditions on the network structures, while still guaranteeing the equations are solved asymptotically.

A note on the MSMAOR method for linear complementarity problems

In this note, without one parameter being greater than the other, a new sufficient condition for convergence of the MSMAOR method for linear complementarity problems is obtained, which is superior to those that appeared in previously published works.

Message-Passing Algorithms: Reparameterizations and Splittings

The max-product algorithm, a local message-passing scheme that attempts to compute the most probable assignment (MAP) of a given probability distribution, has been successfully employed as a method of approximate inference for applications arising in coding theory, computer vision, and machine learning. However, the max-product algorithm is not guaranteed to converge, and if it does, it is not guaranteed to recover the MAP assignment. Alternative convergent message-passing schemes have been proposed to overcome these difficulties. This paper provides a systematic study of such message-passing algorithms that extends the known results by exhibiting new sufficient conditions for convergence to local and/or global optima, providing a combinatorial characterization of these optima based on graph covers, and describing a new convergent and correct message-passing algorithm whose derivation unifies many of the known convergent message-passing algorithms. While convergent and correct message-passing algorithms represent a step forward in the analysis of max-product style message-passing algorithms, the conditions needed to guarantee convergence to a global optimum can be too restrictive in both theory and practice. This limitation of convergent and correct message-passing schemes is characterized by graph covers and illustrated by example.

Characterization of Factor Graph by Mooij's Sufficient Condition for Convergence of the Sum-Product Algorithm

Recently, Mooij et al. proposed new sufficient conditions for convergence of the sum-product algorithm, and it was also shown that if the factor graph is a tree, Mooij's sufficient condition for convergence is always activated. In this letter, we show that the converse of the above statement is also true under some assumption, and that the assumption holds for the sum-product decoding. These newly obtained fact implies that Mooij's sufficient condition for convergence of the sum-product decoding is activated if and only if the factor graph of the a posteriori probability of the transmitted codeword is a tree.

Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant matrices

In this paper, some improvements on Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133] are presented for bounds of the spectral radius of l ω , r , which is the iterative matrix of the generalized AOR (GAOR) method. Subsequently, some new sufficient conditions for convergence of GAOR method will be given, which improve some results of Darvishi and Hessari [On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Appl. Math. Comput. 176 (2006) 128–133].

Convergence and comparison theorems for double splittings of matrices

In this paper, some convergence theorems for the double splitting of a monotone matrix or a Hermitian positive definite matrix are presented. Two comparison theorems for two double splittings of a monotone matrix are obtained. Meanwhile, we establish a new sufficient condition for convergence of the Gauss-Seidel double SOR method for an H-matrix.

A note on convergence conditions of waveform relaxation algorithms for nonlinear differential–algebraic equations

We present and prove a new sufficient condition for convergence of the general waveform relaxation algorithm in the solution of a system of nonlinear differential–algebraic equations. The proof is based on the spectral theory of linear operators. The new condition suggests and we demonstrate that previously published sufficient conditions are unnecessarily restrictive.

Convergence Criteria for Hierarchical Overlapping Coordination of Linearly Constrained Convex Design Problems

Decomposition of multidisciplinary engineering system design problems into smaller subproblems is desirable because it enhances robustness and understanding of the numerical results. Moreover, subproblems can be solved in parallel using the optimization technique most suitable for the underlying mathematical form of the subproblem. Hierarchical overlapping coordination (HOC) is an interesting strategy for solving decomposed problems. It simultaneously uses two or more design problem decompositions, each of them associated with different partitions of the design variables and constraints. Coordination is achieved by the exchange of information between decompositions. This article presents the HOC algorithm and several new sufficient conditions for convergence of the algorithm to the optimum in the case of convex problems with linear constraints. One of these equivalent conditions involves the rank of the constraint matrix that is computationally efficient to verify. Computational results obtained by applying the HOC algorithm to quadratic programming problems of various sizes are included for illustration.

Convergence properties of waveform relaxation circuit simulation methods

A new method for the analysis and understanding of the convergence properties of the waveform relaxation (WR) circuit simulation method is derived. The method describes the convergence properties of WR through a continuous-time linear time-varying (LTV) system convergence equation. From the LTV convergence equation, new sufficient conditions for the convergence of WR are derived that are less restrictive than those of all known previous work. The new sufficient conditions are shown to be valid both for the case of an exact solution of the circuit differential equations and the real-life case of an approximate numerical solution of these equations. An example is given, illustrating the application of the LTV convergence equation and comparing the new sufficient conditions for convergence of the WR method against previous work.

Poisson convergence and poisson processes with applications to random graphs

We give a new sufficient condition for convergence to a Poisson distribution of a sequence of sums of dependent variables. The condition allows each summand to depend strongly on a few of the other variables and to depend weakly on the remaining ones. As a consequence we obtain sufficient conditions for the convergence of point processes, constructed as sets of (weakly) dependent random points in some space S, to a Poisson process. The main applications are to random graph theory. In particular, we solve the problem (proposed by Erdös) of finding the size of the first cycle in a random graph.