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In this paper, we introduce an iteration process to approximate a fixed point of a contractive self-mapping. The comparison theorem indicates that our iteration process is faster than the other existing iteration processes in the literature. We also obtain convergence and stability theorems of this iterative process for a contractive self-mapping. Numerical examples show that our iteration process for approximating a fixed point of a contractive self-mapping is faster than the existing methods. Based on this process, we finally present a new modified Newton-Raphson method for finding the roots of a function and generate some nice polynomiographs.

Machine Learning (ML) solutions are nowadays distributed, according to the so-called server/worker architecture. One server holds the model parameters while several workers train the model. Clearly, such architecture is prone to various types of component failures, which can be all encompassed within the spectrum of a Byzantine behavior. Several approaches have been proposed recently to tolerate Byzantine workers. Yet all require trusting a central parameter server. We initiate in this paper the study of the ``general'' Byzantine-resilient distributed machine learning problem where no individual component is trusted. We show that this problem can be solved in an asynchronous system, despite the presence of $\frac{1}{3}$ Byzantine parameter servers and $\frac{1}{3}$ Byzantine workers (which is optimal). We present a new algorithm, ByzSGD, which solves the general Byzantine-resilient distributed machine learning problem by relying on three major schemes. The first, Scatter/Gather, is a communication scheme whose goal is to bound the maximum drift among models on correct servers. The second, Distributed Median Contraction (DMC), leverages the geometric properties of the median in high dimensional spaces to bring parameters within the correct servers back close to each other, ensuring learning convergence. The third, Minimum-Diameter Averaging (MDA), is a statistically-robust gradient aggregation rule whose goal is to tolerate Byzantine workers. MDA requires loose bound on the variance of non-Byzantine gradient estimates, compared to existing alternatives (e.g., Krum). Interestingly, ByzSGD ensures Byzantine resilience without adding communication rounds (on a normal path), compared to vanilla non-Byzantine alternatives. ByzSGD requires, however, a larger number of messages which, we show, can be reduced if we assume synchrony., This is a merge of arXiv:1905.03853 and arXiv:1911.07537; arXiv:1911.07537 will be retracted

With China's economy entering the stage of high-quality development, manufacturing energy carbon emission efficiency has become the focus of academic attention. It is of great significance to study the convergence of manufacturing energy carbon emission efficiency for realizing high-quality development of manufacturing in China. Based on the panel data of China's manufacturing sub-sectors, this paper measures and analyzes the evolution trend of manufacturing energy carbon emission and its efficiency. On this basis, this paper uses the coefficient of variation and convergence model to test the convergence of manufacturing energy carbon emission efficiency. The results show that China's manufacturing energy carbon emissions and its efficiency demonstrate an increasing trend. Coal was the main source of manufacturing energy carbon emissions. The manufacturing energy carbon emission efficiency does not have σ convergence, but has [Formula: see text] convergence, and its convergence has industry heterogeneity. The manufacturing energy carbon emission efficiency exits scale effect and technology effect, but not the effect of opening to the outside world and institutional effect, and its effect exists industry heterogeneity. By reducing carbon emissions, adopting differentiated policies, adjusting the industry scale, and enhancing the industry technology intensity, China's manufacturing can improve the energy carbon emission efficiency and promote high-quality economic development.

Economic growth in the information age is no longer a stage driven by unipolarity. It has entered a multi-polar driving stage characterized by integration, fusion, and integrated development on a larger scale between regions, and the trend of group competition with urban agglomerations as carriers has become increasingly obvious. This paper improves the neural network algorithm based on the needs of industrial economic integration in the digital age, and proposes an industry convergence analysis model based on the improved neural network algorithm. Moreover, this article combines industry models to analyze actual needs and constructs an industry convergence analysis model based on improved neural networks, and analyzes the integration of different industries. In addition, this article conducts experiments through multiple sets of data, and combines the neural network model of this article to conduct research. Through experimental research, we know that the model constructed in this paper can play an important role in the analysis of industry convergence.

The traditional least squares (LS) and gradient descent (GD) algorithms can estimate the parameters of the regression models. They can be inefficient when the models have complex structures: (1) the unknown parameters in the information vector make the algorithm be impossible to update the parameters; (2) the zigzagging nature of the gradient descent algorithm and the complex structures lead to slow convergence rates; and (3) the step-size and derivative function calculations may be unsolvable for complex nonlinear models. This paper proposes two kinds of algorithms for exponential nonlinear models. The first is the two-stage algorithm, which decomposes the complex model into a linear part and a nonlinear part, where the linear part is estimated using the LS algorithm and the nonlinear part is identified based on the GD algorithm. The second is the particle swarm optimization algorithm which can simultaneously obtain all the parameters. To increase the convergence rates, the Aitken method is also introduced. The simulation results demonstrate the effectiveness of the proposed algorithms.

The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate relative error tolerance. Second, an inexact non-monotone line search scheme is employed to compute a step size which defines the next iteration. It is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual scaled gradient projection method employing monotone line searches., Comment: 30 pages, 14 figures

It is shown that the solution to the boundary - initial value problem for a Kelvin–Voigt fluid of order one depends continuously upon the Kelvin–Voigt parameters, the viscosity, and the viscoelastic coefficients. Convergence of a solution is also shown.

Melanoma is one of the main causes of cancer-related deaths. The development of new computational methods as an important tool for assisting doctors can lead to early diagnosis and effectively reduce mortality. In this work, we propose a convolutional neural network architecture for melanoma diagnosis inspired by ensemble learning and genetic algorithms. The architecture is designed by a genetic algorithm that finds optimal members of the ensemble. Additionally, the abstract features of all models are merged and, as a result, additional prediction capabilities are obtained. The diagnosis is achieved by combining all individual predictions. In this manner, the training process is implicitly regularized, showing better convergence, mitigating the overfitting of the model, and improving the generalization performance. The aim is to find the models that best contribute to the ensemble. The proposed approach also leverages data augmentation, transfer learning, and a segmentation algorithm. The segmentation can be performed without training and with a central processing unit, thus avoiding a significant amount of computational power, while maintaining its competitive performance. To evaluate the proposal, an extensive experimental study was conducted on sixteen skin image datasets, where state-of-the-art models were significantly outperformed. This study corroborated that genetic algorithms can be employed to effectively find suitable architectures for the diagnosis of melanoma, achieving in overall 11% and 13% better prediction performances compared to the closest model in dermoscopic and non-dermoscopic images, respectively. Finally, the proposal was implemented in a web application in order to assist dermatologists and it can be consulted at http://skinensemble.com.

The moving asymptotes function is widely applied as sequential explicit convex approximation in topology optimization, due to the controllable conservatism and convergence. In virtue of these advantages, it is supposed that the efficiency of optimality criteria method will become higher if constraint functions are approximated by moving asymptotes function instead of linear function. This work presents an optimality criteria method for topology optimization under multiple constraints where the constraint functions are approximated by moving asymptotes function. The dual feasibility condition is customarily adopted to establish explicit update scheme of topological variable, where hypothesis of active topological variable set is avoided, in the case that gradient of objective function is positive. The complementary slackness condition and primal feasibility condition are combined into a simplified dual programming to solve for the Lagrange multipliers, where hypothesis of active constraint set is avoided. Three benchmark examples under multiple displacement, stress, compliance or eigenfrequency constraints are solved by the presented optimality criteria method, the results are compared to the method of moving asymptotes and the optimality criteria method with linear constraint approximation.

In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known "fast iterative shrinkage-thresholding algorithm" (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.

The differential evolution (DE) algorithm is an efficient random search algorithm based on swarm intelligence for solving optimization problems. It has the advantages of easy implementation, fast convergence, strong optimization ability and good robustness. However, the performance of DE is very sensitive to the design of different operators and the setting of control parameters. To solve these key problems, this paper proposes an improved self-adaptive differential evolution algorithm with a shuffled frog-leaping strategy (SFSADE). It innovatively incorporates the idea of the shuffled frog-leaping algorithm into DE, and at the same time, it cleverly introduces a new strategy of classification mutation, and also designs a new adaptive adjustment mechanism for control parameters. In addition, we have carried out a large number of simulation experiments on the 25 benchmark functions of CEC 2005 and two nonparametric statistical tests to comprehensively evaluate the performance of SFSADE. Finally, the results of simulation experiments and nonparametric statistical tests show that SFSADE is very effective in improving DE, and significantly improves the overall diversity of the population in the process of dynamic evolution. Compared with other advanced DE variants, its global search speed and optimization performance also has strong competitiveness.

We consider the finite element (FE) approximation of the two dimensional shallow water equations (SWE) by considering discretizations in which both space and time are established using a stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to establish stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We establish a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptive mesh refinements on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to some commonly applied benchmarks for the SWE.

Redundancy manipulators need favorable redundancy resolution to obtain suitable control actions to guarantee accurate kinematic control. Among numerous kinematic control applications, some specific tasks such as minimally invasive manipulation/surgery require the distal link of a manipulator to translate along such fixed point. Such a point is known as remote center of motion (RCM) to constrain motion planning and kinematic control of manipulators. Recurrent neural network (RNN) which possesses parallel processing ability, is a powerful alternative and has achieved success in conventional redundancy resolution and kinematic control with physical constraints of joint limits. However, up to now, there still is few related works on the RNNs for redundancy resolution and kinematic control of manipulators with RCM constraints considered yet. In this paper, for the first time, an RNN-based approach with a simplified neural network architecture is proposed to solve the redundancy resolution issue with RCM constraints, with a new and general dynamic optimization formulation containing the RCM constraints investigated. Theoretical results analyze and convergence properties of the proposed simplified RNN for redundancy resolution of manipulators with RCM constraints. Simulation results further demonstrate the efficiency of the proposed method in end-effector path tracking control under RCM constraints based on a redundant manipulator.

Long short-term memory (LSTM) neural network has been widely studied and applied in the real world. To obtain the LSTM neural network with better accuracy and more appropriate structure, the hybrid coding particle swarm optimization (HCPSO) algorithm is proposed. Firstly, the hybrid coding scheme is developed to represent the weights and structure of LSTM neural network, simultaneously. Then, the novel update mechanism is proposed to adjust the position of particles. Meanwhile, the discrete update strategy (DUS) and adaptive nonlinear moderate random search strategy (ANMRS) are proposed to enhance the convergence and global search capability of HCPSO, respectively. Finally, the effectiveness of HCPSO is demonstrated by multiple numerical examples. The experiment results show that the proposed HCPSO algorithm is more competitive in optimizing LSTM neural networks than other algorithms.

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

The application of Levy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Levy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.

This article develops a method based on the generalized Gegenbauer–Humbert wavelets in concert with their operational matrices of fractional integration to deal with the fractional partial differential equations and find the approximate solutions of it. The goal is to show that the proposed method is appropriate for boundary and initial-boundary problems even though it is generalized form. The convergence of the method under study is investigated. The numerical results gained by the proposed method are considered and compared with other methods, to establish the effectiveness and accuracy.

The shuffled frog leaping algorithm is easily trapped into local optimum and has the low optimization accuracy when it is used to optimize the complex functions problems. To overcome the above shortcomings, a shuffled frog leaping algorithm with contraction factor was proposed. By introducing acceleration factors c1 and c2, the ability of worst individual to learn from best individual within the submemeplexes or global best individual of the entire population was improved and the convergence rate of algorithm was accelerated. Under inserting the contraction factor χ, the convergence of algorithm was ensured. After performing local optimization of the self-learning operator on the worst individual, and taking full advantage of the useful information in the worst individuals, the self-learning ability of the individual and the optimization accuracy of the algorithm were improved. Simulation results illustrated that the enhanced algorithm performed better optimization performance than basic SFLA and other improved SFLAs. Finally, the proposed algorithm was used to optimize five problems of the mechanical design, and its validity and practicability were verified.

In this study, a closed-loop $${D^\alpha }$$ -type iterative learning control (ILC) with a proportional D-type iterative learning updating law for the initial shift is applied to nonlinear conformable fractional order system. First, the system with the initial shift is introduced. Then, fractional order ILC (FOILC) frameworks that experience the initial shift problem for the path tracking of nonlinear conformable fractional order systems are addressed. Moreover, the sufficient condition for the convergence of tracking errors is obtained in the time domain by introducing $$\lambda $$ -norm and Holder’s inequality. Lastly, numerical examples are provided to illustrate the effectiveness of the proposed methods.

How to maintain a good balance between convergence and diversity is particularly important for the performance of the many-objective evolutionary algorithms. Especially, the many-objective optimization problem is a complicated Pareto front, the many-objective evolutionary algorithm can easily converge to a narrow of the Pareto front. An efficient environmental selection and normalization method are proposed to address this issue. The maximum angle selection method based on vector angle is used to enhance the diversity of the population. The maximum angle rule selects the solution as reference vector can work well on complicated Pareto front. A penalty-based adaptive vector distribution selection criterion is adopted to balance convergence and diversity of the solutions. As the evolution process progresses, the new normalization method dynamically adjusts the implementation of the normalization. The experimental results show that new algorithm obtains 30 best results out of 80 test problems compared with other five many-objective evolutionary algorithms. A large number of experiments show that the proposed algorithm has better performance, when dealing with numerous many-objective optimization problems with regular and irregular Pareto Fronts.

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

In recent years, many multi-objective evolutionary algorithms have been proposed to solve many-objective optimization problems with regular Pareto front. These algorithms have shown good performance in balancing convergence and diversity. However, in the high-dimensional objective space, the non-dominated solutions increases exponentially as the number of objectives increases. The metrics to evaluate algorithm performance are also computationally intensive. In particular, solving the many-objective optimization problem of the irregular Pareto front faces great challenges. Moreover, many-objective evolutionary algorithms, do not easily show their convergence and diversity through visualization, as multi-objective evolutionary algorithms do. To address these problems, a many-objective optimization algorithm based on decomposition and hierarchical clustering selection is proposed in this paper. First, a set of uniformly distributed reference vectors divides non-dominanted individuals into different sub-populations, and then candidate solutions are selected based on the aggregation function values in the sub-populations. Second, a set of adaptive reference vectors is used to rank the dominant individuals in the population and retain promising candidate solutions. Third, a hierarchical clustering selection strategy is used to enable solutions with good convergence to be selected. Finally, a diversity maintenance strategy is used to remove solutions with poor diversity. The experimental results show that the proposed algorithm EA-DAH has advantages over other comparative algorithms in many-objective optimization problems with irregular Pareto fronts.

Grey Wolf Optimizer (GWO) has been proposed recently. As GWO has superior performance, it has been employed to solve various numerical and engineering issues. However, it easily traps into stagnation when solving complex and multimodal problems. GWO mainly searches around the top three wolves and assigns the same weights to them, deteriorating the convergence and exploration. A reinforced exploitation and exploration GWO (REEGWO) is developed. In the proposed REEGWO algorithm, the top three wolves are given different weights on the basis of their knowledge about the location of the prey. Then, a random search based on the tournament selection is used to enhance the exploration. A well-designed mechanism is developed to balance exploration and exploitation. The experimental results have proved that REEGWO is perfect among GWO and its four recently top variants. Then, the proposed REEGWO is compared with the latest heuristic algorithms and their latest variants. The results have shown that REEGWO is competitive. Four real-world applications are also solved by six algorithms, and results have further validated the efficiency of REEGWO.

Detection and recovery of noise corrupted signals using adaptive filters are becoming popular due to their application in several fields, including communication and biomedical. Adaptive filters broadly utilize the LMS algorithm since it has low computational complexity and is robust in behavior. However, it fails to achieve a faster speed of convergence and minimum steady-state MSE simultaneously. Therefore, the objective is to estimate the noise-free signal faster with improved accuracy and obtain a lower steady-state MSE with a higher convergence speed. This paper introduces a multi-stage adaptive filtering model wherein the noisy signal is analyzed through a cascade of stages. The proposed signal de-noising scheme utilizes the automatic selection of stages to be cascaded such that the steady-state MSE is minimum. Further, to enhance convergence speed, the LMS adaptive filter’s step-size is automatically adjusted at each stage. Hence, by controlling the number of stages to be cascaded automatically and utilizing a different step-size for each stage, we obtain a high convergence speed and minimum steady-state MSE. The proposed filter structure is tested for signal de-noising application to assess its performance concerning MSE, Signal-to-Noise Ratio (SNR), Average Noise Reduction (ANR), and convergence speed. The results attained have shown that the proposed filter structure provides remarkable performance improvement at different input noise levels. Further, the proposed Improved Variable-Stage (IVS) cascaded adaptive filter model employs LMS adaptive algorithm, hence offering a cost-effective and straightforward hardware implementation of ANC.

Infrared images generally suffer from low resolutions, which restricts the further data mining of these images. Super-resolution (SR) technology effectively improves the spatial resolution of the infrared image without changing the existing hardware imaging equipment. Therefore, it is a promising computational imaging approach with resource-limited edge devices. In this paper, an adaptive-threshold nonlocal means (NLM)-based SR algorithm is proposed. Specifically, an image quality assessment index of infrared SR results is designed and introduced into the NLM reconstruction algorithm. On the one hand, it is used as a threshold to determine the iterative convergence condition of the algorithm; on the other hand, it is used as an evaluation standard to select the best reconstructed HR image among multiple output results. A GPU acceleration strategy is also proposed to ensure the high efficiency of the edge computing process for reducing the computational time. Experimental results demonstrate that the algorithm realizes the adaptive iteration of the NLM-based SR of the infrared images, and it significantly improves the geometric structures and detail recognition of the original input LR images. The sawtooth and ringing effects of the algorithm are less, and its objective evaluation indexes are also significantly improved compared with those of other SR algorithms.

The aim of this paper is to prove the existence and uniqueness of common fixed points for a pair of $$(\psi -\varphi )$$ ( ψ - φ ) -weak contractive self-maps in the setting of b-metric spaces satisfying the minimal requirement of weakly compatibility, and other weak commuting properties as compatibility, R-weakly commuting and R-weakly commuting of types ($$A_T$$ A T ), ($$A_S$$ A S ) and ($$A_P$$ A P ). Also, we will analyze the convergence and stability of the Jungck-Noor iterative scheme for this class of pairs of mappings on b-metric spaces endowed with a convex structure.

This paper develops a weak-label-based global and local multi-view multi-label learning with three-way clustering (WL-GLMVML-ATC) to solve multi-view multi-label data sets and exploit more authentic global and local label correlations of both the whole data set and each view simultaneously. Different from the traditional learning methods, WL-GLMVML-ATC pays more attention to the solutions of weak-label cases and uncertain relationships of clusters with the usage of Universum and active three-way clustering. According to Universum notion, even though the size of labeled instances is much more smaller than the unlabeled ones, the useful sample information can still be enhanced. Through the active three-way clustering strategy, the belongingness of instances to a cluster depend on the probabilities of uncertain instances belonging to core regions. This strategy brings a more authentic local label correlation since many traditional methods suppose that instances and the corresponding clusters always exhibit certain relationships such as belong-to definitely and not belong-to definitely. This hypothesis is not ubiquitous in real-world applications. According to the experiments, we can see WL-GLMVML-ATC (1) achieves a better performance, be superior to the classical multi-view learning methods and multi-label learning methods in statistical, advances the development of these learning methods in final; (2) won’t add too much running time; (3) has a good convergence and ability to process multi-view multi-label data sets.

The growing scale of marine exploration requires high-resolution underwater localization, which necessitates cooperation among underwater network nodes, considering the channel complexity and power efficiency. In this paper, we proposed factor graph weight particles aided distributed underwater nodes cooperative positioning algorithm (WP-DUCP). It capitalized on the factor graph and sum-product algorithm to decompose the global optimization to the product of several local optimization functions. Combined with the Gaussian parameters to construct the weighted particles and to realize the belief transfer, it shows low complexity and high efficiency, suitable to the energy-restricted and communication distance-limited underwater networks. In terms of convergence, localization resolution, and computation complexity, we conducted the simulation and real-test with comparison to the existing co-localization methods. The results verified a higher resolution of the proposed method with no extra computation burden.

The stability of integrators dealing with high order Differential Algebraic Equations (DAEs) is a major issue. The usual procedures give rise to instabilities that are not predicted by the usual linear analysis, rendering the common checks (developed for ODEs) unusable. The appearance of these difficult-to-explain and unexpected problems leads to methods that arise heavy numerical damping for avoiding them. This has the undesired consequences of lack of convergence of the methods, along with a need of smaller stepsizes. In this paper a new approach is presented. The algorithm presented here allows us to avoid the interference of the constraints in the integration, thus allowing the linear criteria to be applied. In order to do so, the integrator is applied to a set of instantaneous minimal coordinates that are obtained through the application of the null space. The new approach can be utilized along with any integration method. Some experiments using the Newmark method have been carried out, which validate the methodology and also show that the method behaves in a predictable way if one considers linear stability criteria.

The four-dimensional (4D) printing technology, as a combination of additive manufacturing and smart materials, has attracted increasing research interest in recent years. The bilayer structures printed with smart materials using this technology can realize complicated deformation under some special stimuli due to the material properties. The deformation prediction of bilayer structures can make the design process more rapid and thus is of great importance. However, the previous works on deformation prediction of bilayer structures rarely study the complicated deformations or the influence of the printing process on deformation. Thus, this paper proposes a new method to predict the complicated deformations of temperature-sensitive 4D printed bilayer structures, in particular to the bilayer structures based on temperature-driven shape-memory polymers (SMPs) and fabricated using the fused deposition modeling (FDM) technology. The programming process to the material during printing is revealed and considered in the simulation model. Simulation results are compared with experiments to verify the validity of the method. The advantages of this method are stable convergence and high efficiency, as the three-dimensional (3D) problem is converted to a two-dimensional (2D) problem. The simulation parameters in the model can be further associated with the printing parameters, which shows good application prospect in 4D printed bilayer structure design.

This paper introduces an interactive framework to guide decision-makers in a multi-criteria supplier selection process. State-of-the-art multi-criteria methods for supplier selection elicit the decision-maker’s preferences among the criteria by processing pre-collected data from different stakeholders. We propose a different approach where the preferences are elicited through an active learning loop. At each step, the framework optimally solves a combinatorial problem multiple times with different weights assigned to the objectives. Afterwards, a pair of solutions among those computed is selected using a particular query selection strategy, and the decision-maker expresses a preference between them. These two steps are repeated until a specific stopping criterion is satisfied. We also introduce two novel fast query selection strategies, and we compare them with a myopically optimal query selection strategy. Computational experiments on a large set of randomly generated instances are used to examine the performance of our query selection strategies, showing a better computation time and similar performance in terms of the number of queries taken to achieve convergence. Our experimental results also show the usability of the framework for real-world problems with respect to the execution time and the number of loops needed to achieve convergence.

In this study, two fractional-based stochastic gradient (FSG) algorithms for time-delayed auto-regressive exogenous (ARX) models are proposed. By combining momentum and adaptive methods, a momentum-based FSG and an adaptive-based FSG algorithms are developed. These two FSG algorithms have faster convergence rates when compared with the stochastic gradient algorithm. The mechanism of the convergence is proved in theory. Furthermore, two simulated examples are presented to illustrate the efficiency of the new proposed algorithms.

Swarm intelligence (SI) is part of artificial intelligence that is based on the collective behavior of particles in a self-organized and decentralized intelligent systems which is inspired from either the human or animal social behavior. SI has a great involvement in solving optimization problems. As optimization-based systems are complicated, so powerful decentralized algorithms supported by SI are used to solve such problems. Intelligent algorithms of SI can solve the challenging issues of optimization due to their different properties; thus, there is an increased demand to enhance the performance and propose new and novel SI algorithms. This paper presents a novel stochastic swarm intelligence algorithm called success history intelligent optimizer (SHIO). It offers a solution for single-objective optimization problems by proposing a new exploration and exploitation movement strategy based on the three best solutions found in the search space to create a new movement vector, where each best solution is stored in the memory and subtracted from the average of the three best solutions found so far during the optimization process. The proposed SHIO ensures the efficiency of search space exploration and use. In order to confirm SHIO performance, several performance measurements (search history, trajectory and convergence curves) have been tested and SHIO was used to solve (23) single-objective optimization benchmarking functions. These functions have been classified to unimodal, multimodal and multimodal fixed. Various metrics such as mean, standard deviation, minimum and maximum have been utilized, and quantitative findings have been recorded. Further, trajectory and search history of the qualitative result were visualized. The results of test functions and performance metrics demonstrate that the proposed algorithm can explore various search area locations, make use of potential search space locations while optimizing, avoid local optimism and converge to the global best efficiently. SHIO delivers highly competitive and superior results in the evaluated unimodal and multimodal benchmarks over the compared algorithms. Note that SHIO algorithm source code is available on http://hussamfakhouri.org and open for public use.

The measurement of traditional workpiece geometry is based on the measurement results of points, lines, surfaces, balls, and other elements as objects. The best measurement area in the measurement space of the on-machine measurement system for different typical measurement objects is determined to achieve a high-precision, efficient measurement. This study is oriented to the circular surface measurement object, establishes the integrated error model and circular surface measurement error model of the on-machine measurement system, and analyzes the circular surface measurement error distribution law in the designated measurement space. Beetle antennae search particle swarm optimization algorithm (BAS-PSO) is proposed to solve the optimal measurement area for the established theoretical objective model of the optimal measurement area. The optimization effect is compared between BAS-PSO and the ACO, PSO, FOA, SA-PSO (simulated annealing particle swarm optimization), and IA-PSO (immune particle swarm optimization) algorithms. The experiment of determining the best measurement area of the on-machine measurement system based on circular surface measurement is conducted, simulated, and calculated with the circular surface measurement error model. The comparative analysis of the best measurement area is obtained by algorithm optimization. Experimental results show that the BAS-PSO algorithm is not caught in the local optimum, its convergence speed and stability are better than those of the five other algorithms, and it is more suitable for solving the best measurement area of the circular surface measurement. The circular surface error measurement results and simulation calculations overlap with the best measurement area obtained by the algorithm optimization. The space of the best area is 356.061 mm ≤ X ≤ 365.061 mm, −109.727 mm ≤ Y ≤ −100.727 mm, and −263 mm ≤ Z ≤ −253 mm. The measured maximum error minimum value of this area is 5.3 μm. The minimum value of the maximum error obtained by the simulation calculation of the circular surface measurement is 4.1 μm, and the minimum value of the maximum error obtained by the algorithm optimization is 8.8 μm. The validity and practicability of the proposed method for determining the best measurement area of the three-axis machine tool on-machine measurement system are verified.

In this paper, Lie classical approach is utilized for the symmetry reduction of time-fractional variable coefficient Caudrey–Dodd–Gibbon–Sawada–Kotera equation. The obtained symmetries and Erd $$\acute{e}$$ lyi–Kober fractional differential operator are used to reduce the original nonlinear partial differential equation into nonlinear ordinary differential equation. The generalized Noether operator and new conservation theorem are exploited to obtain conservation laws of the governing equation. The power series solution is also derived for the considered equation. The obtained power series solution is investigated for the convergence and the obtained power series solution is convergent.

With renewable energy currently making the headlines, photovoltaic technology has shown significant potential as one of the best energy sources. It thus becomes necessary to predict the performance of photovoltaic systems by modeling it accurately and optimally. We propose a new hybrid algorithm for extracting PV cell parameters to improve their performance and efficiency when subjected to temperature variations. A metaheuristic approach is combined with an analytical approach to improve the accuracy and robustness. We call this approach improved differential evolution (IDE). The performance of the proposed method is evaluated for selected cell data. For validation, several analyses and comparisons are made with other methods, and the results illustrate the accuracy and precision of IDE. The proposed technique can estimate the parameters in an optimal way at any temperature, together with high convergence speed and a short simulation time.

An arithmetic optimizer algorithm (AOA) is hybridized with slime mould algorithm (SMA) to address the issue of less internal memory and slow convergence at local minima which is termed as HAOASMA. Lens opposition-based learning strategy is also integrated with the hybrid algorithm which enhances the population diversity of the hybrid optimizer to accelerate the convergence. The local best ( $$P_{{{\text{best}}}} )$$ and global best ( $$g_{{{\text{best}}}} )$$ of SMA initializes the AOA’s search process. The $$P_{{{\text{best}}}}$$ obtained from AOA again initializes the SMA to further exploit the search space. In this way, the developed hybrid algorithm utilizes the exploitation and exploration capabilities of SMA and AOA, respectively. The developed HAOASMA has been compared on twenty-three benchmark functions at different dimensions with basic SMA, AOA and six renowned meta-heuristic algorithms. The HAOASMA has also been applied to classical engineering design problems. The performance of HAOASMA is significantly superior compared to basic SMA, AOA and other meta-heuristic algorithms.

To solve the problems of premature convergence and easily falling into local optimum, a whale optimization algorithm based on dynamic pinhole imaging and adaptive strategy is proposed in this paper. In the exploitation phase, the dynamic pinhole imaging strategy allows the whale population to approach the optimal solution faster, thereby accelerating the convergence speed of the algorithm. In the exploration phase, adaptive inertial weights based on dynamic boundaries and dimensions can enrich the diversity of the population and balance the algorithm’s exploitation and exploration capabilities. The local mutation mechanism can adjust the search range of the algorithm dynamically. The improved algorithm has been extensively tested in 20 well-known benchmark functions and four complex constrained engineering optimization problems, and compared with the ones of other improved algorithms presented in literatures. The test results show that the improved algorithm has faster convergence speed and higher convergence accuracy and can effectively jump out of the local optimum.

The finite-volume theory is an equilibrium-based approach characterized by the equilibrium equations' local satisfaction and the imposition of continuity conditions in a surface-averaged sense through the subvolume interfaces. Previous investigations include stress and displacement fields convergence and computational cost, showing the approach's efficiency, especially in heterogeneous materials and structures. However, those investigations did not include an energy analysis, which is especially important in compliance minimization problems. As the finite element method, energy-based approaches impose energy balance, which guarantees a monotonic energy convergence. This contribution addresses a numerical investigation of the main mechanical energy aspects involving the generalized finite-volume theory for continuum elastic structures in quasi-static analyses. The obtained results are verified with analytical and finite element-based analyses, showing the monotonic energy convergence for the three versions of the finite-volume theory and the energy balance's satisfaction for the higher-order versions when a sufficiently refined mesh is employed.

The time-fractional Black-Scholes model governing European options in which the temporal derivative is focused on the Caputo fractional derivative with $$0

A convergent algorithm for nonnegative matrix factorization with orthogonality constraints imposed on both factors is proposed in this paper. This factorization concept was first introduced by Ding et al. with intent to further improve clustering capability of NMF. However, as the original algorithm was developed based on multiplicative update rules, the convergence of the algorithm cannot be guaranteed. In this paper, we utilize the technique presented in our previous work to develop the algorithm and prove that it converges to a stationary point inside the solution space., Comment: arXiv admin note: substantial text overlap with arXiv:1010.5290

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier–Stokes equations on the torus subject to a random perturbation converges in $$L^2(\varOmega )$$ , and describe the rate of convergence for an $$H^1$$ -valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the $$L^2(\varOmega )$$ -convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott–Vogelius mixed elements and for an additive noise, the convergence is polynomial.

Fuzzy grey cognitive maps (FGCMs) have been widely adopted to support cause-and-effect decision-making under uncertainty. However, capturing the information for the initial state vector and the relationship matrix from various specialists can be cumbersome, which may affect the convergence of FGCMs or cause them to reach a chaotic state. To address this issue, this paper presents a novel group decision approach based on the combination of grey clustering (GC) and fuzzy grey cognitive maps for assessing causal relationships in uncertain environments. The main contribution consists in applying GC as a mean to obtain the initial state vector from the relationship matrix data. This halves the required inputs to the users, reducing uncertainty in the computational model. Also, the proposition brings a method for aggregating the linguistic judgements of multiple decision-makers when assessing causal relationships. The proposition also differs from others in the literature by providing results with lower imprecision levels, named greyness, and lower number of required iterations for the FGCM-based system to converge. A real application was conducted in a technology start-up to test the approach to practical implications. Results allowed the identification of important elements regarding the company’s profile and performance, aiding prioritization and enabling the development of action plans. This paper also includes a comparison of other representative models with the proposed approach, which led to more accurate results. Hence, this study addresses the need for new alternatives to improve the reliability and convergence of FGCM-based systems. Finally, suggestions for future applications are proposed.

Deep nonnegative matrix factorization (DMF) is a particularly useful technique for learning data representation in low-dimensional space. To further obtain the complex hidden information and keep the geometrical structures of the high-dimensional data, we propose a novel deep matrix factorization model with the graph regularization (called DGsnMF). For solving the model with multi-variables, we design a forward–backward splitting scheme. After that, the convergence analysis is attached to the proposed algorithm and it is proved to converge to a critical point. Empirical experiments on benchmark datasets show that the proposed method is superior to the compared methods.

In the present paper, we consider the semilocal convergence issue of two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. As applications of our convergence results, a nonsymmetric algebraic Riccati equation arising from transport theory and a two-dimensional nonlinear convection-diffusion equation are provided.

The advent of multifactorial optimization (MFO) has made a wind of change in intelligence computation in general and specifically in evolutionary computing. Based on the implicit parallelism of population-based search, MFO optimizes different problems simultaneously and entirely. However, the randomness of knowledge transfers raises the question of how to diminish harmful interactions among tasks for more effective transfers. In recent years, many proposals have been devised to handle this paradigm and improve existing algorithms. Notwithstanding the diversity in their concept, there are few efforts to solve many-task optimization (MaTO) that contains beyond three tasks. In light of this reason, this paper proposes two algorithms named SA-MFEA and LSA-MFEA for MaTO. Instead of utilizing fixed parameters, SA-MFEA and LSA-MFEA adapt the probability of random mating parameter to reduce negative transfers based on the historical memory of successful rmp. Besides, LSA-MFEA is capable of enhancing the exploitation by linear population size reduction. To examine the efficiency of the two proposed algorithms, experiments on various many-task benchmark problems and comparison with several state-of-the-art algorithms have been conducted. The results demonstrated that SA-MFEA and LSA-MFEA are competitive in terms of quality of solutions, convergence trend, and computation time.

In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence.

We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results are provided under loose assumptions, making our theoretical results a practical tool for analyzing a family of various reweighted algorithms. The effectiveness and efficiency of our proposed formulation and the algorithms are demonstrated in numerical experiments on various sparse optimization problems.

This paper considers the adaptive finite-time tracking control problem for a class of switched nonlinear systems. Based on neural networks, the influence of unknown nonlinear functions is offset. A performance function is proposed to guarantee the transient performance of the tracking error. Moreover, a novel-type Lyapunov function is constructed that relaxes the constraints on virtual control functions. It is shown by Lyapunov stability analysis that the finite-time convergence and zero-error tracking performance can be ensured simultaneously. Finally, the effectiveness of the proposed method is verified by a simulation example.