Convergence Properties Research Articles

Approximation properties by Schurer type q-Kantorovich–Stancu shifted knots operators

We design the Schurer type Kantorovich–Stancu operators by using shifted knots in the quantum calculus. We obtain the convergence and other related approximation properties of these operators. We discuss the degree of convergence of our operators by applying the modulus of continuity. In addition, we give some basic direct theorems and obtain the approximation in Lipschitz spaces.

An Iterative Method for the Approximation of Common Fixed Points of Two Mappings: Application to Fractal Functions

This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides sufficient conditions for the strong convergence of the sequence generated by the iterative scheme to a common fixed point of two operators. The method is illustrated with some examples of application. The procedure is used to approach a common solution of two Fredholm integral equations of the second kind. In the second part of the article, the existence of a fractal function coming from two different Read–Bajraktarević operators is proved. Afterwards, a study of the approximation of fixed points of a fractal convolution of operators is performed, in the framework of Lebesgue or Bochner spaces.

Performance iterative learning control on nonlinear systems for tracking varying-scale and morphologically similar targets

This paper proposes a novel three-dimensional (3D) performance iterative learning control (PILC) scheme on nonlinear systems for tracking varying-scale and morphologically similar targets. The condition and performance weighted K-nearest neighbour (KNN) and the historical database update algorithms are developed to match the historical control inputs closest to the current operating conditions, and further realise the learning process along the performance domain. The control inputs of the PILC scheme can be updated by utilising the control information obtained from the time, iteration and performance dimensions. Also, the convergence property of the proposed framework is proved via the contraction mapping principle. Finally, compared with the existing ILC works, the illustrative simulation and experimental results are presented to illustrate that the PILC scheme has better tracking performance and robustness, smaller initial error, and faster convergence speed.

Formulation of Variable Step Block Backward Differentiation Formula for the Computation of Nonlinear Fuzzy Differential Equations

The research paper formulates a variable step block backward differentiation formula (VSBBDF) for solving nonlinear fuzzy differential equations (FDEs). Developed to address uncertainties within differential equations by using fuzzy environments, VSBBDF offers a flexible approach to solve equations with triangular fuzzy numbers. This method incorporates a dynamic step-size selection, allowing it to adapt to changes and reduce computational costs while maintaining accuracy. The paper further discusses the stability and convergence properties of VSBBDF, demonstrating that it is both zero-stable and of high accuracy. The results obtained highlight the method's computational efficiency and reliability, particularly when compared with other existing methods for solving nonlinear FDEs.

Recent Trends in Regional Disparity in India: A State-level Club Convergence Study Using Night-time Light Data

It is a consensus in academic discourse that regional disparity in India is not converging. Nonetheless, some studies argue that conclusions regarding regional convergence are sensitive to the measures employed to define regional disparities. In this study, we use night-time light radiance data from 2012 to 2021 to illustrate the converging pattern of regional disparity in India, specifically employing state-level VIIRS night-time light data for this purpose. Our analysis concludes that there is no regional convergence among all sample states in India. However, the club convergence hypothesis is relevant here, as 12 states form two clubs and exhibit convergence properties. In contrast, 13 major states in India show no signs of convergence. Furthermore, we identify three states that are diverging. Thus, based on night-time light data, we conclude that Indian states have not shown a convergence pattern recently, and regional disparity remains a significant issue in the Indian economy. This finding supports the existing literature in the Indian context, where regional convergence is absent among Indian states. JEL: O40, O47, R11, R12

Iterative methods for projection onto the ℓ p quasi-norm ball

In this paper, we focus on computing the projection onto the ℓ p quasi-norm ball. Motivated by the existing IRBP algorithm, which tackles ℓ p quasi-norm ball projection through projections onto a sequence of simple and tractable weighted ℓ 1 norm balls, we propose an innovative and effective iteratively reweighting approach for solving the projection problem. Specifically, our method centres on an ϵ -approximation that yields better approximation accuracy. We propose an innovative non-smooth, yet continuously differentiable approximation to the ℓ p quasi-norm function. Leveraging the concavity of our approximate model, we developed a novel variant of the iterative reweighted ℓ 1 norm ball projection algorithm. Through rigorous analysis, we demonstrated the global convergence properties of our proposed numerical approach. Numerical studies demonstrates the superior computational efficiency of our proposed algorithm. Moreover, our solution method can be further leveraged within current iterative algorithms designed to solve ℓ p quasi-norm ball constrained optimization problems, especially in scenarios where rapid convergence is of utmost importance.

Line-search based optimization using function approximations with tunable accuracy

This paper develops a line-search algorithm that uses objective function models with tunable accuracy to solve smooth optimization problems with convex constraints. The evaluation of objective function and its gradient is potentially computationally expensive, but it is assumed that one can construct effective, computationally inexpensive models. This paper specifies how these models can be used to generate new iterates. At each iteration, the model has to satisfy function error and relative gradient error tolerances determined by the algorithm based on its progress. Moreover, a bound for the model error is used to explore regions where the model is sufficiently accurate. The algorithm has the same first-order global convergence properties as standard line-search methods, but only uses the models and the model error bounds. The algorithm is applied to problems where the evaluation of the objective requires the solution of a large-scale system of nonlinear equations. The models are constructed from reduced order models of this system. Numerical results for partial differential equation constrained optimization problems show the benefits of the proposed algorithm.

Memoryless quasi-Newton-type methods via some weak secant relations for large-scale unconstrained optimization

This article presents two variants of memoryless quasi-Newton methods with backtracking line search for large-scale unconstrained minimization. These updating methods are derived by means of a least-change updating strategy subjected to some weaker form of secant relation obtained by projecting the secant equation onto the search direction. In such a setting, the search direction can be computed without the need of calculation and storage of matrices. We establish the convergence properties for these methods, and their performance is tested on a large set of test functions by comparing with standard methods of similar computational cost and storage requirement. Our numerical results indicate that significant improvement has been achieved with respect to iteration counts and number of function evaluations.

Fedeval: Defending Against Lazybone Attack via Multi-dimension Evaluation in Federated Learning

Federated learning (FL) has become a prominent paradigm for collaborative model training while ensuring data privacy. However, in resource-constrained environments, such as the Internet of Things (IoT), FL faces a distinct challenge from Lazybone attackers, who compromise system performance by providing low-quality data or conducting minimal local training to reduce their computational burden. In this paper, we propose Fedeval, a novel multi-dimensional evaluation framework designed to defend against Lazybone attacks. Fedeval leverages a server-side base validation dataset and a base model to assess the quality and relevance of client contributions through gradient inversion, and it compares client-uploaded gradients with an honest baseline to detect training inconsistencies. By assigning adaptive importance scores based on client contributions, Fedeval enhances the robustness of FL by mitigating the impact of non-contributing participants. We also provide a theoretical analysis of Fedeval’s convergence properties and validate its effectiveness through extensive experiments on four datasets and two attack scenarios. Our results demonstrate that Fedeval significantly accelerates convergence and improves accuracy by up to 13% compared to traditional methods.

Approximation with Chlodowsky variant of Kantorovich-Stancu-operators employing associated λ-polynomials

This article explores a Chlodowsky-type extension of the Kantorovich variant of Szász operators based on the associated λ-polynomials. The convergence properties of these operators are established by employing the universal Korovkin-type property, and the order of approximation is determined using the classical modulus of continuity and the Ditzian-Totik modulus of continuity. Additionally, the αβ-statistical convergence properties of the operators are derived. Additional approximation outcomes are also formulated within the context of the polynomial weighted space. Finally, the theoretical results are supported by numerical and graphical examples for different parameters.

Covariance-Modulated Optimal Transport and Gradient Flows

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.

Experimental investigation of heralded Gaussification of phase-randomized coherent states of light

Probabilistic heralded Gaussification of quantum states of light is an important ingredient of protocols for distillation of continuous variable entanglement and squeezing. An elementary step of the heralded Gaussification protocol consists of interference of two copies of the state at a balanced beam splitter, followed by conditioning on the outcome of a suitable Gaussian quantum measurement on one output mode. When iterated, the protocol either converges to a Gaussian state or diverges. Here, we report on experimental investigation of the convergence properties of iterative heralded Gaussification. We experimentally implement two iterations of the protocol, which requires simultaneous processing of four copies of the input state. We utilize the phase-randomized coherent states as the input states, which greatly facilitates the experiment because these states can be generated deterministically, and their mean photon number can easily be tuned. We comprehensively characterize the input and partially Gaussified states by balanced homodyne detection and quantum state tomography. Our experimental results are in good agreement with theoretical predictions and they provide new insights into the convergence properties of heralded quantum Gaussification.

Distributed Coordination D-Stabilization in Cyclic Pursuit Formations of Dynamical Multi-Agent Systems

In this study, the cyclic pursuit formation stabilization problem in target-enclosing operations by multiple homogeneous dynamic agents is investigated. To this end, a Lyapunov D-stability problem is first formulated to cover the transient performance requirements for multi-agent systems. Then, a simple diagrammatic Lyapunov D-stability criterion for cyclic pursuit formation is derived. The formation control scheme combined with a cyclic-pursuit-based distributed online path generator satisfying this condition guarantees both the required transient performance and global convergence properties with theoretical rigor. It is shown that the maximization of the connectivity gain in a cyclic-pursuit-based online path generator can be reduced to an optimization problem subject to linear matrix inequality constraints derived using the generalized Kalman-Yakubovich–Popov lemma. This approach provides a permissible range of connectivity gain, which not only guarantees global formation stability/convergence properties but also satisfies the required performance specification. Several numerical examples are provided to confirm the effectiveness of the proposed method.

Terminal Synergetic Control with the Dragonfly Algorithm for Zoonotic Visceral Leishmaniasis Eradication

Visceral Leishmaniasis (VL) is a prevalent vector-borne disease that affects both human and animal populations in subtropical and tropical regions, contributing to a substantial mortality rate. Establishing efficient control policies is crucial to eradicating the VL epidemic. The VL epidemic system, containing reservoirs, vectors, and human populations, can be accurately modeled through differential equations. Managing the VL epidemic under multiple control policies can be considered a high-order nonlinear feedback control challenge. This study explores the application of Terminal Synergetic Control (TSC) to eradicate Zoonotic Visceral Leishmaniasis (ZVL). Notably, Synergetic Control (SC) is one of the suitable feedback control methods for manipulating high-order nonlinear systems, providing practical control inputs because of their chattering-free behavior. Additionally, the convergence properties of the control system can be enhanced through terminal attraction. Optimization of control parameters within the system is achieved through the integration of control mechanisms by the Dragonfly Algorithm (DA). The results demonstrate that the multiple control policies synthesized by the TSC method effectively regulate subpopulations in alignment with the specified control objectives. Furthermore, the enhanced convergence rate achieved by the TSC method, in comparison to the SC method, serves as evidence of TSC's effectiveness in guiding the dynamics of ZVL epidemic eradication. This research underscores the potential of the TSC method, utilizing optimal control parameters provided by the DA, to achieve targeted outcomes with improved convergence properties.

Stability of the F∗ Algorithm on Strong Pseudocontractive Mapping and Its Application

This paper investigates the stability of the F∗ iterative algorithm applied to strongly pseudocontractive mappings within the context of uniformly convex Banach spaces. The study leverages both analytic and numerical methods to demonstrate the convergence and stability of the algorithm. In comparison to previous works, where weak-contraction mappings were utilized, the strongly pseudocontractive mappings used in this study preserve the convergence property, exhibit greater stability, and have broader applicability in optimization and fixed point theory. Additionally, this work shows that the type of mapping employed converges faster than those in earlier studies. The results are applied to a mixed-type Volterra–Fredholm nonlinear integral equation, and numerical examples are provided to validate the theoretical findings. Key contributions of this work include the following: (i) the use of strongly pseudocontractive mappings, which offer a more stable and efficient convergence rate compared to weak-contraction mappings; (ii) the application of the F∗ algorithm to a wider range of problems; and (iii) the proposal of future directions for improving convergence rates and exploring the algorithm’s behavior in Hilbert and reflexive Banach spaces.

Viscosity implicit midpoint scheme for enriched nonexpansive mappings

This article proposes and analyses a viscosity scheme for an enriched nonexpansive mapping. The scheme is incorporated with the implicit midpoint rule of stiff differential equations. We deduce some convergence properties of the scheme and establish that a sequence generated therefrom converges strongly to a fixed point of an enriched nonexpansive mapping provided such a point exists. Furthermore, we provide some examples of the implementation of the schemes with respect to certain enriched mappings and show the numerical pattern of the scheme.

Optimization schemes on manifolds for structured matrices with fixed eigenvalues

AbstractSeveral manifold optimization schemes are presented and analyzed for solving a specialized inverse structured symmetric matrix problem with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered. The rest of the entries are free, some of them preferably away from zero. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem is related to the problem of determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our optimization schemes are based on considering the eigenvector matrix as the only unknown and iteratively moving on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose Riemannian gradient-type methods combined with two different well-known retractions, and with two well-known constrained optimization strategies: penalization and augmented Lagrangian. We also present a block alternating technique that takes advantage of a proper separation of variables. Convergence properties of the penalty alternating approach are established. Finally, we present initial numerical results to demonstrate the effectiveness of our proposals.

Hybrid feature selection of microarray prostate cancer diagnostic system

DNA microarray prostate cancer diagnosis systems are widely used, and hybrid feature selection methods are applied to select optimal features to address the high dimensionality of the dataset. This work proposes a new hybrid feature selection method, namely the relief-F (RF)-genetic algorithm (GA) with support vector machine (SVM) classification method. The aim is to evaluate the performance of the proposed method in terms of accuracy, computation time, and the number of selected features. The method is implemented using Python in PyCharm and is evaluated on a DNA microarray prostate cancer. The outcome of this work is a performance comparison table for the proposed methods on the dataset. The performance of GA, particle swarm optimization (PSO), and whale optimization algorithm (WOA) is compared in terms of accuracy, computation time, and the number of selected features. Results show that GA has the highest average accuracy (91.17%) compared to PSO (90.52%) and WOA (85.74%). GA outperforms PSO and WOA due to its superior convergence properties and better alignment with complex problems.

Reconstructing seismic data using the accelerated linearized Bregman method and iterative soft thresholding algorithm

Seismic data reconstruction plays a crucial role in seismic exploration and imaging, facilitating accurate subsurface characterization and geological interpretation. In this study, we propose a novel approach for reconstructing seismic data using the accelerated linearized Bregman method (ALBM) in conjunction with the iterative soft thresholding algorithm (ISTA). This combined method, referred to as ALBM+ISTA, aims to exploit the complementary strengths of ALBM’s accelerated convergence properties and ISTA’s sparsity-promoting capabilities. The proposed approach is evaluated through theoretical simulations and case studies, wherein specific quantitative metrics such as signal-to-noise ratio (SNR), structural similarity index, root mean square error, peak signal-to-noise ratio, and edge preservation index are employed to assess reconstruction quality and accuracy. Results demonstrate that ALBM+ISTA offers improved performance in terms of noise suppression, structural fidelity, and edge preservation compared to existing reconstruction techniques. Furthermore, the method exhibits adaptability to diverse seismic acquisition geometries and noise levels, making it a promising tool for enhancing seismic data processing in various geological science applications.

Curious Explorer: a Provable Exploration Strategy in Policy Learning.

A coverage assumption is critical with policy gradient methods, because while the objective function is insensitive to updates in unlikely states, the agent may need improvements in those states to reach a nearly optimal payoff. However, this assumption can be unfeasible in certain environments, for instance in online learning, or when restarts are possible only from a fixed initial state. In these cases, classical policy gradient algorithms like REINFORCE can have poor convergence properties and sample efficiency. Curious Explorer is an iterative state space pure exploration strategy improving coverage of any restart distribution ρ. Using ρ and intrinsic rewards, Curious Explorer produces a sequence of policies, each one more exploratory than the previous one, and outputs a restart distribution with coverage based on the state visitation distribution of the exploratory policies. This paper main results are a theoretical upper bound on how often an optimal policy visits poorly visited states, and a bound on the error of the return obtained by REINFORCE without any coverage assumption. Finally, we conduct ablation studies with REINFORCE and TRPO in two hard-exploration tasks, to support the claim that Curious Explorer can improve the performance of very different policy gradient algorithms.