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

Tabu search (TS) is a metaheuristic that guides a local heuristic search procedure to explore the solution space beyond local optimality. It has achieved widespread successes in solving practical optimization problems. This paper proposes the stochastic TS strategy for discrete optimization and makes an investigation of its global convergence. The strategy considered introduces the Metropolis criterion and simulated annealing process into a general framework of TS. It has been proved that the strategy converges asymptotically to global optimal solutions, and satisfies the necessary and sufficient conditions for global asymptotic convergence. Furthermore, it produces a higher convergent rate than the simulated annealing algorithm.

Simulated annealing (SA) is a generic optimization method that is quite popular because of its ease of implementation and its global convergence properties. However, SA is widely reported to converge very slowly, and it is common practice to allow extra freedom in its design at the expense of losing global convergence guarantees. A natural way to increase the flexibility of SA is to allow the objective function and the communication mechanism to be temperature-dependent, the idea being to gradually reveal the complexity of the optimization problem and to increase the mixing rate at low temperatures. We call this general class of annealing processes stochastic continuation (SC). In the first part of this paper, we introduce SC starting from SA, and we derive simple sufficient conditions for the global convergence of SC. Our main result is interesting in two respects: first, the conditions for global convergence are surprisingly weak--in particular, they do not involve the variations of the objective function with temperature--and second, exponential cooling makes it possible to be arbitrarily close to the best possible convergence speed exponent of SA. The second part is devoted to the application of SC to the problem of producing aesthetically pleasing drawings of undirected graphs. We consider the objective function defined by Kamada and Kawai (Inf Process Lett 31(1):7---15, 1989), which measures the quality of a drawing as a weighted sum of squared differences between Euclidean and graph-theoretic inter-vertex distances. Our experiments show that SC outperforms SA with optimal communication setting both in terms of minimizing the objective function and in terms of standard aesthetic criteria.

Large-scale renewable energy bases increasingly employ automatic voltage control (AVC) to coordinate heterogeneous reactive-power resources. The resulting voltage regulation process inherently involves sampling, communication delay, and nonlinear device characteristics, which may induce nontraditional voltage oscillations and stability degradation that cannot be adequately captured by conventional continuous-time or small-signal analysis. This paper proposes a discrete-time nonlinear voltage stability analysis framework for renewable energy bases with multi-reactive-power-resource coupling under AVC-based coordinated control. The voltage regulation dynamics are formulated as a discrete-time nonlinear closed-loop system by incorporating sampled AVC actions, delayed voltage feedback, and nonlinear voltage–reactive-power coupling. An incremental system representation is constructed, and a strong-contraction-based stability criterion is derived using sector-bounded nonlinearity descriptions and linear matrix inequalities, providing a sufficient condition for global voltage convergence without local linearization. Extensive numerical studies are conducted on a representative renewable energy base with parallel and series coupling topologies. A total of 2916 randomized configurations are evaluated. The proposed criterion achieves consistency rates exceeding 96% for the parallel topology and 99% for the series topology when compared with time-domain simulations, while the probability of dangerous misjudgment remains below 1%. Scenario-based simulations further demonstrate that coupling topology plays a critical role in shaping voltage stability behaviors, and state-space analysis further supports the observed stability behaviors. These results indicate that nonlinear strong contraction offers an effective and practical stability notion for AVC-based voltage regulation in renewable energy bases.

Intergenerational mobility and macroeconomic history dependence

In this paper, we consider continuous-time semidecentralized dynamics for the equilibrium computation in a class of aggregative games. Specifically, we propose a scheme where decentralized projected-gradient dynamics are driven by an integral control law. To prove global exponential convergence of the proposed dynamics to an aggregative equilibrium, we adopt a quadratic Lyapunov function argument. We derive a sufficient condition for global convergence that we position within the recent literature on aggregative games, and in particular we show that it improves on established results.

This contribution discusses the global convergence behavior of iterative algorithms solving nonlinear approximation problems via a sequence of linear approximation problems. Essentially there are two sufficient conditions for global convergence: 1) The algorithm should have only critical points as accumulation points. 2) The parametrization should imply that the parameters of refined approximations have accumulation points. For some important nonlinear families of approximating functions it is proved that the above conditions hold, provided that the parametrization is restricted to suitable subfamilies.

In optimization in with m nonlinear equality constraints, we study the local convergence of reduced quasi-Newton methods, in which the updated matrix is of order n−m Furthermore, we give necessary and sufficient conditions for superlinear convergence (in one step) and we introduce a device to globalize the local algorithm, It consists in determining a step along an arc in order to decrease an exact penalty function and we give conditions so that asymptotically the step-size will be equal to one.

We investigate local convergence of an SQP method for an optimal control problem governed by a parabolic equation with nonlinear boundary condition. Sufficient conditions for local quadratic convergence of the method are discussed.

Modeling and analysis of a coupled SIS bi-virus model

In the first part of our theses we give sufficient conditions for the absolute convergence of the double Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to double Fourier series. In the second part we give sufficient conditions for the absolute convergence of the double Walsh-Fourier series of a function. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity and s-bounded fluctuation.

The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

Let [special characters omitted] be function of [special characters omitted], 0 [special characters omitted], and [special characters omitted], that takes m x m nonnegative matrix values and such that [special characters omitted] for all [special characters omitted]) and all [special characters omitted]. Denote by [special characters omitted] the space of N-vector valued functions continuous on the closed unit disc in [special characters omitted] and analytic inside the disc whose Taylor series expansion about 0 has real coefficients. We consider the following problem: [special characters omitted]where [special characters omitted] is a subset of [special characters omitted] that may depend on given continuous N-vector valued functions [special characters omitted] whose entries have real Fourier coefficients, on given continuous scalar valued functions [special characters omitted] with real Fourier coefficients, and on given real constants cl. The set [special characters omitted] we consider takes one of the following forms:[special characters omitted]We establish optimality conditions for each of these three cases that must be satisfied by a solution of the (CONSTRAINED-OPT) problem. These optimality conditions can be used for finding good candidates to solutions of the (CONSTRAINED-OPT), and we discuss algorithms for finding such candidates for [special characters omitted] as in (1) and (2). Numerical examples are presented. We also consider the problem of finding a spectral factorization of a given, possibly low rank, positive semidefinite matrix valued function on the unit circle of the complex plane. We derive operator equations that must be satisfied by solutions to the spectral factorization problem for either the low or the full rank cases. The equations we derive can be solved numerically using Newton's method. We prove that sufficient conditions for local quadratic convergence of the Newton's method are satisfied. Numerical examples are given. Finally, we develop an implementation of a high level algorithm of Peller-Young for finding superoptimal approximants to a given matrix valued continuous function. Our implementation uses spectral factorization as well as other tools that we developed for this purpose. Also a numerical example is presented.

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

A number of results exist in the literature for singularly perturbed differential equations without turning points. In particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence. This guarantees that the schemes model the boundary layers well. We wish to examine whether these schemes will also be uniformly convergent, if the equation has turning points. To this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes. We show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one. We confirm these theoretical results by numerical calculations.

In the present note, we extend some univariate uniform approximation results by means of Lagrange interpolating polynomials [Ivan, M., Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca (2004)] to the bivariate case. It is well known that generally, in the univariate case, the sequence of Lagrange interpolation polynomials does’t converges to the approximated function. This fact was first observed by G. Faber (see [9]), which constructed an example when the sequence of Lagrange interpolation polynomials diverges. The result of G. Faber was more generalized by I. Muntean (see [12]). M. Ivan established first sufficient conditions for the uniform convergence of the sequence of Lagrange interpolation polynomials associated to a univariate real valued function. First, we represent the remainder term of bivariate Lagrange interpolation formula in terms of bivariate divided difference. Using this representation we establish sufficient conditions for the uniform convergence of the sequence of bivariate Lagrange interpolation polynomials to the approximated function.