Conditions For Convergence Research Articles

Adaptive event-triggered [formula omitted] consensus secondary control for islanded microgrids under multiple time delays

In this paper, an adaptive event-triggered H∞ consensus control scheme is proposed for the secondary layer of AC MGs. The controller design problem of topological communication network with external disturbance, multiple delay, and communication burden is considered. Therefore, using the Lyapunov-Krasovskii theory, we obtain sufficient conditions for the robust consensus convergence of multi-agents under event-triggered and multiple time delays. Because traditional event-triggered often use fixed thresholds, it is challenging to balance reducing the communication burden and maintaining excellent system performance. Therefore, we design an adaptive event-triggered method that adaptively adjusts the trigger threshold according to the consistency error. Finally, a simulation system is established. The proposed control strategy has been validated through simulation results.

A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space

The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects.

Convergence and complexity guarantees for a wide class of descent algorithms in nonconvex multi-objective optimization

We address conditions for global convergence and worst-case complexity bounds of descent algorithms in nonconvex multi-objective optimization. Specifically, we define the concept of steepest-descent-related directions. We consider iterative algorithms taking steps along such directions, selecting the stepsize according to a standard Armijo-type rule. We prove that methods fitting this framework automatically enjoy global convergence properties. Moreover, we show that a slightly stricter property, satisfied by most known algorithms, guarantees the same complexity bound of O(ϵ−2) as the steepest descent method.

Physical meaning of conditionally convergent series: the calculation of the Madelung constant

Conditionally convergent series are infinite series whose result depends on the order of the sum. One of the most famous examples of conditionally convergent series of interest in Physics is the calculation of Madelung’s constant α in ionic crystals. The appearance of this type of series is quite disturbing to students and often causes misunderstandings. In this work we analyze the physical meaning of the conditional convergence from a pedagogical point of view. The problem is posed using a toy model of ionic crystal in which the lattice sums can be calculated explicitly for various forms of expansion of the crystal about a central core. It is seen directly how the Coulomb series does not converge to α when there are charge accumulations on the surfaces. Therefore, it becomes clear what the appropriate strategy should be when choosing the order of summation to arrive at the correct value of α.

Model-free compensation learning control of asymmetric hysteretic systems with initial state learning

In this article, a model-free compensation learning control scheme is presented for asymmetric hysteretic systems to achieve high-precision output tracking, where the effects of asymmetric input hysteresis nonlinearities are denoted by the asymmetric Prandtl–Ishlinskii model (APIM). In the presented control scheme, the feedforward iterative learning technique is generalized to the control design of asymmetric hysteretic systems. To improve the robustness and dynamic performance of the existing feedforward learning control for hysteretic systems, both the feedback action and the differential term are utilized to design a discrete-time PD-P open-closed-loop learning control law for simultaneously compensating the asymmetric input hysteresis nonlinearities and the linear dynamics effects without constructing compensators based on their models. The initial state error of such a system is also considered, a modified initial state learning strategy is proposed to ensure the initial state error tends to a prescribed level with the increasing of iterations. By fully analyzing the properties of the APIM, the convergence conditions with respect to the input error, the state error, and the tracking error along the iteration domain are given. The simulation and experimental results are provided to demonstrate strong robustness and excellent tracking accuracy with the proposed model-free compensation learning control scheme.

A many-objective evolutionary algorithm under diversity-first selection based framework

Many-objective optimization problems (MaOPs) have attracted wide attention. However, most solving methods prioritize the convergence or take the convergence and diversity into account. This causes the difficulty in balancing convergence and diversity of the population still remains, especially for these problems with irregular Pareto fronts. To this end, we propose a many-objective evolutionary algorithm under diversity-first selection based framework (DSFMO), which first emphasizes the diversity and then considers the convergence. Thus, DSFMO can drive individuals to approach the Pareto front from various directions, and further achieve the balance between convergence and diversity of the population. In this framework, we first construct a candidate pool by selecting these individuals with small crowded degree (i.e., good diversity), in which the crowded degree is evaluated by the global diversity measure based on the objective transferring. Then, the individual in the constructed candidate pool that has the best convergence is selected to enter the next generation, where the convergence is evaluated by the conditional convergence measure that considers the importance of boundary individuals and evolution status. The above two steps are repeatedly executed until the next generation population is formed. In addition, a space projection assisted mating selection is developed to select high-quality parents for variation and further enhance the exploration and exploitation ability of DSFMO. Experiments on 88 test instances have shown that DSFMO can perform best on most test instances. In other words, DSFMO outperforms its seven competitors in balancing convergence and diversity of the population.

Research on Group Behavior Modeling and Individual Interaction Modes with Informed Leaders

This study investigates coordinated behaviors and the underlying collective intelligence in biological groups, particularly those led by informed leaders. By establishing new convergence condition based on experiments involving real biological groups, this research introduces the concept of a volitional term and heterogeneous networks, constructing a coupled-force Cucker–Smale model with informed leaders. Incorporating informed leaders into the leader-follower group model enables a more accurate representation of biological group behaviors. The paper then extracts the Flock Leadership Hierarchy Network (FLH), a model reflecting real biological interactions. Employing time slicing and rolling time windows, the study methodically analyzes group behavior stages, using volatility and convergence time as metrics to examine the relationship between group consistency and interactions. Comparative experiments show the FLH network’s superior performance. The Kolmogorov-Smirnov test demonstrates that the FLH network conforms to a power-law distribution, a prevalent law in nature. This result further illuminates the crucial role that power-law distribution plays in the evolutionary processes of biological communities. This study offers new perspectives on the evolution of biological groups, contributing to our understanding of the behaviors of both natural and artificial systems, such as animal migration and autonomous drone operations.

Kinematic parameters and Raychaudhuri equation in the Kantowski-Sachs model

In this work we present a review of the calculations of homogeneous and anisotropic cosmological models, particularly the Kantowski-Sachs cosmological model. As an innovative contribution, we calculate an exact solution to the field equations with a general expression for the material content of the Universe. We obtain the expressions for energy density in this general case. We also study the case involving a scalar field. We investigate the evolution dynamics of the model, encompassing the Hubble, density, and deceleration parameters. We calculate the kinematic parameters associated with the model, namely: the scalar expansion Θ, the shear scalar σ, the rotation parameter ω and the parameter R α β U α U β , also known as Raychaudhuri scalar or convergence condition, all these parameters expressed in the Raychaudhuri equation.

Well-posedness for the surface quasi-geostrophic front equation

We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal. 3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.

Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on R n \mathbb {R}^n . While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on R n \mathbb {R}^n .

Решение оптимизационной задачи оценивания моделей полносвязной линейной регрессии

<p>This article is devoted to the problem of estimating fully connected linear regression models using the maximum likelihood method. Previously, a special numerical method was developed for this purpose, based on solving a nonlinear system using the method of simple iterations. At the same time, the issues of choosing initial approximations and fulfilling sufficient conditions for convergence were not studied. This article proposes a new method for solving the optimization problem of estimating fully connected regressions, similar to the method of estimating orthogonal regressions. It has been proven that, with equal error variances of interconnected variables, estimates of b-parameters of fully connected regression are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse covariance matrix. And if the ratios of the error variances of the variables are equal to the ratios of the variances of the variables, then the b-parameter estimates are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse correlation matrix, multiplied by the specific ratios of the standard deviations of the variables. A numerical experiment was carried out to confirm the correctness of the developed mathematical apparatus. The proposed method for solving the optimization problem of estimating fully connected regressions can be effectively used when solving problems of constructing multiple fully connected linear regressions.</p>

Mixture generalized minimum error entropy-based distributed lattice Kalman filter

The Gaussian kernel function-based Minimum Error Entropy (MEE) criterion is effective for special types non-Gaussian noise. However, non-Gaussian noise distributions and shapes are diverse in practice, the traditional MEE methods are difficult to fit non-Gaussian effectively due to the shape parameters of MEE cannot be adjusted. In this paper, the Mixture Generalized Minimum Error Entropy (MGMEE) criterion is proposed by a mixture generalized Gaussian kernel function. Then, a new Mixture Generalized Minimum Error Entropy-based Distributed Lattice Kalman Filter (MGMEE-DLKF) is proposed for multi-sensor nonlinear systems with non-Gaussian noise. The complexity analysis and convergence condition of proposed MGMEE-DLKF algorithm are derived. In the end, the target tracking simulations are verified for systems with mixture Gaussian noise, Rayleigh distribution noise and α−stable distribution noise. The simulation results demonstrate that the proposed filter has the smallest root mean square error.

Trigonometric series and the permutation sign convergence condition

Trigonometric series and the permutation sign convergence condition

Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function

Abstract There are several problems in physics, such as kinetic energy equation, wave equation, anomalous diffusion process, and viscoelasticity that are described well in the fractional differential equation form. Therefore, the solutions with elementary solution method cannot be solved and described deliberately with detailed physics of the problems, so these problems are solved with the help of special operators such as Mittag–Leffler (M–L) functions equipped with Riemann–Liouville (R–L) fractional operators. Hence, keeping in view the above-mentioned problems in physics in the current study, the generalized properties are derived M–L functions connected with R–L fractional operators that are investigated in the generalized form. These extended special operators will be used for the solutions of generalized kinetic energy equation. The M–L function is a fundamental special function with a wide range of applications in mathematics, physics, engineering, and various scientific disciplines. Ayub et al. gave the definition of newly extended M–L ( p , s , k ) \left(p,s,k) function. Also, they gave its convergence condition and found several results relevant to that. The purpose of this study is to investigate newly extended M–L function and study its elementary properties and integral transforms such as Whittaker transform and fractional Fourier transform. The R–L fractional operator is a fundamental concept in fractional calculus, a branch of mathematics that generalizes differentiation and integration to non-integer orders. In this study, we discuss the relation of M–L ( p , s , k ) \left(p,s,k) -function and R–L fractional operators. In some cases, fractional calculus is used to describe kinetic energy equations, particularly in systems where fractional derivatives are more appropriate than classical integer-order derivatives. The M–L function can appear as a solution or as a part of the solution to these fractional kinetic energy equations. Also, we gave the generalization of kinetic energy equation and its solution in terms of newly extended M–L function.

Resilient distributed optimization for cyber–physical systems under adversarial environments: An event-based method

This work presents a resilient distributed optimization algorithm based on the event-triggering mechanism for cyber–physical systems (CPSs) to optimize an average of convex cost functions corresponding to multiple agents under adversarial environments. Two attack scenarios, including the f-total (each agent is affected by at most f malicious agents in the whole network) and the f-local (each agent is affected by at most f malicious agents in its in-neighbor set) attacks are considered. Subsequently, the convergence conditions under these two attack scenarios are provided, respectively, both of which guarantee that the state values of benign agents converge to a bounded error range. The optimality conditions are also presented by theoretical analysis, which guarantee that the state values of benign agents converge to a safety interval constructed by local optimal values under certain graph conditions, despite the misbehavior of malicious agents. In addition, four numerical examples are presented to show the effectiveness and superiority of the event-triggering resilient distributed optimization (RDO-E) algorithm. Compared to existing resilient algorithms, the proposed method achieves resilient distributed optimization with higher accuracy and less demanding communication overheads. Finally, by applying the proposed method to the multi-microgrid system, a resilient economic dispatch problem (REDP) is successfully solved, which validates the practical viability of the RDO-E algorithm.

Computationally efficient robust adaptive filtering algorithm based on improved minimum error entropy criterion with fiducial points

Recently, there has been a strong interest in the minimum error entropy (MEE) criterion derived from information theoretic learning, which is effective in dealing with the multimodal non-Gaussian noise case. However, the kernel function is shift invariant resulting in the MEE criterion being insensitive to the error location. An existing solution is to combine the maximum correntropy (MC) with MEE criteria, leading to the MEE criterion with fiducial points (MEEF). Nevertheless, the algorithms based on the MEEF criterion usually require higher computational complexity. To remedy this problem, an improved MEEF (IMEEF) criterion is devised, aiming to avoid repetitive calculations of the aposteriori error, and an adaptive filtering algorithm based on gradient descent (GD) method is proposed, namely, GD-based IMEEF (IMEEF-GD) algorithm. In addition, we provide the convergence condition in terms of mean sense, along with an analysis of the steady-state and transient behaviors of IMEEF-GD in the mean-square sense. Its computational complexity is also analyzed. Simulation results demonstrate that the computational requirement of our algorithm does not vary significantly with the error sample number and the derived theoretical model is highly consistent with the learning curve. Ultimately, we employ the IMEEF-GD algorithm in tasks such as system identification, wind signal magnitude prediction, temperature prediction, and acoustic echo cancellation (AEC) to validate the effectiveness of the IMEEF-GD algorithm.

On Construction of Subgame Perfect Nash Equilibria in Stackelberg Games

Identifying a Subgame Perfect Nash Equilibrium (SPNE) of a two-player Stackelberg game could be not a manageable task, especially when the players have a continuum of actions and the follower’s best reply correspondence is not single-valued. Aim of the paper is to investigate the issue of construction of SPNEs in Stackelberg games by exploiting perturbations of both the action sets and the payoff functions of the leader and the follower. To achieve the goal, since the limit of SPNEs of perturbed games is not necessarily an SPNE of the original game even for classical perturbations, we prove under nonrestrictive convergence conditions how to produce an SPNE starting from a sequence of SPNEs of general perturbed games. This result allows to describe a procedure to find SPNEs that can accommodate various types of perturbations. More precisely, under mild assumptions on the data of the original game, we show that a large class of perturbed games (including, for example, perturbation approaches relying on the Tikhonov and entropic regularizations or motivated by altruistic and antagonistic behaviors) satisfies the convergence conditions for constructing an SPNE. The specific SPNE selections associated to such a class, together with their possible behavioral interpretations, are discussed and an illustrative example is provided.

A revisit to classical and quantum aspects of Raychaudhuri equation and possible resolution of singularity

In this review, we provide a concrete overview of the Raychaudhuri equation, Focusing Theorem and convergence conditions in a plethora of backgrounds and discuss the consequences. We also present various classical and quantum approaches suggested in the literature that could potentially mitigate the initial big-bang singularity and the black-hole singularity.

Exploring trends and determinants of basic childhood vaccination coverage: Empirical evidence over 41 years.

Vaccination is widely considered to be one of the most important prevention measures as a health strategy. This paper examines trends in basic childhood vaccination coverage and which country and time-dependent determinants may have influenced childhood immunization rates (1-dose BCG, 1- and 3-dose DTP (diphtheria, tetanus, pertussis), 1-dose measles, and 3-dose polio) between 1980 and 2020 across 94 countries. We identify economic, inequality, demographic, health, education, labor market, environmental, and political stability factors of immunization. To do this, we use data from the annual WHO and United Nations International Children's Emergency Fund (UNICEF) coverage estimates. The empirical analysis consists of generalized estimating equation models to assess relationships between immunization rates and socioeconomic factors. Additionally, we follow the Barro and Sala-i-Martín approach to identify conditional convergence. Our findings show the strongest positive statistically significant association between immunization rates and GDP per capita, as well as births attended by skilled health staff. Moreover, our research demonstrates conditional convergence, indicating that countries converge towards different steady states. The present study brings new insights to investigating the determinants of childhood vaccination coverage and provides significant implications for health policies.

Phase-only beampattern synthesis for maximizing mainlobe gain via Riemannian Newton method

Beampattern synthesis with phase-only weights is extensively utilized in phased-array systems due to the virtue of low-cost and high power efficiency. It is always a non-convex optimization problem having constant modulus constraints (CMCs) and the performance significantly depends on the employed solvers. This paper discusses the phase-only beampattern synthesis for maximizing the mainlobe gain with the constraints on the mainlobe ripple and the relative sidelobe level. As CMC is equivalent to a complex circle manifold, the proposed beampattern synthesis is considered as a manifold optimization problem, and then decomposed into several subproblems under the framework of manifold alternating direction method of multipliers (ADMM). In particular, the subproblem for updating phase-only weights is efficiently tackled by a second-order Riemannian Newton method, which has faster convergence speed and no requirement on line search compared to widely-used first-order Riemannian manifold methods. The explicit convergence condition is derived and the computational complexity is also analyzed. Numerical results show that the proposed method has better performance on the mainlobe gain and sidelobe control accuracy, or less computational cost compared to several representative approaches, such as convex relaxation, standard ADMM, Riemannian gradient descent-based ADMM and Riemannian conjugate gradient methods.