In this article, the variational generalized Nash equilibrium (vGNE) seeking problem for general monotonic game with multiple coupling constraints involving dynamical players is explored. Specifically, a distributed vGNE-seeking neural network (vGSNN) with a feedback controller is designed based on high-pass filter, which efficiently transforms players' high-order dynamics into equivalent second-order ones. To further relax the requirement on parameter predesign, we propose a controller that uses adaptive weights to replace the traditional fixed gains, which realizes the full distribution of the vGSNN. Furthermore, to enhance the robustness of the vGSNN against disturbances, a novel sliding-mode controller is incorporated to ensure finite-time disturbance rejection while maintaining the full distribution of the vGSNN. Finally, an uncrewed aerial vehicle (UAV) swarm game is put forward to verify the effectiveness of the vGSNNs.

As deep learning models are increasingly deployed in high-risk applications, robust defenses against adversarial attacks and reliable performance guarantees become paramount. Moreover, accuracy alone does not provide sufficient assurance or reliable uncertainty estimates for these models. This study advances adversarial training by leveraging principles from Conformal Prediction. Specifically, we develop an adversarial attack method, termed OPSA (OPtimal Size Attack), designed to reduce the efficiency of conformal prediction at any significance level by maximizing model uncertainty without requiring coverage guarantees. Correspondingly, we introduce OPSA-AT (Adversarial Training), a defense strategy that integrates OPSA within a novel conformal training paradigm. Experimental evaluations demonstrate that our OPSA attack method induces greater uncertainty compared to baseline approaches for various defenses. Conversely, our OPSA-AT defensive model significantly enhances robustness not only against OPSA but also other adversarial attacks, and maintains reliable prediction. Our findings highlight the effectiveness of this integrated approach for developing trustworthy and resilient deep learning models for safety-critical domains. Our code is available at https://github.com/bjbbbb/Enhancing-Adversarial-Robustness-with-Conformal-Prediction.

Although many product providers deem user contributions (e.g. online reviews) important, providers often struggle to obtain them, i.e. most users are lurkers who are reluctant to post reviews. This study is conducted to understand better how to delurk users by investigating the role of user stickiness in review posting behaviour. By employing a large-scale dataset from TapTap, a Chinese mobile game community, and conducting a multimethod investigation, this study found that sticky users with a product are more likely to engage in review posting behaviour related to the product. Also, this positive stickiness-post effect varies for some user and product features: (a) the positive relationship between user stickiness and posting behaviour will be strengthened as user expertise rises, (b) the positive stickiness-post relationship will be alleviated when products are collaborative-consuming products (vs. private-consuming products), (c) and the positive stickiness effect on review posting behaviour will be stronger when product providers are small-scale (vs. large-scale). These findings provide a comprehensive understanding of biases in lurking/posting behaviour related to user stickiness and help product providers gain insights into delurking users and gathering user intelligence.

The concept of proper equilibrium was established as a strict refinement of perfect equilibrium. This establishment has significantly advanced the development of game theory and its applications. Nonetheless, it remains a challenging problem to compute such an equilibrium. This paper develops a differentiable path-following method with a compact formulation to compute a proper equilibrium. The method incorporates square-root-barrier terms into payoff functions with an extra variable and constitutes a square-root-barrier game. As a result of this barrier game, we acquire a smooth path to a proper equilibrium. To further reduce the computational burden, we present a compact formulation of an ε-proper equilibrium with a polynomial number of variables and equations. Numerical results show that the differentiable path-following method is numerically stable and efficient. Moreover, by relaxing the requirements of proper equilibrium and imposing Selten’s perfection, we come up with the notion of perfect d-proper equilibrium, which approximates a proper equilibrium and is less costly to compute. Numerical examples demonstrate that even when d is rather large, a perfect d-proper equilibrium remains to be a proper equilibrium. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms-Continuous. Funding: This work was partially supported by General Research Fund (GRF) CityU 11306821 of Hong Kong SAR Government. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0148 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0148 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In this paper, we investigate a neural adaptive formation control problem for underactuated unmanned surface vehicles (USVs). Considering the limitation of communication distance and the security of formation systems, collision-free and connectivity maintenance are guaranteed by defining a prescribed-time tuning function and proper error transformation. Furthermore, a new nonlinear first-order filter, solving the complexity problem, is designed to promote the system performance. Subsequently, neural networks (NNs) are used to approximate USVs’ dynamics and their transient performance is improved by prediction error. By blending prediction errors and neural approximation, it is guaranteed the general external disturbances and approximation errors are compensated via constructed disturbance observers (DOs), simultaneously. Meanwhile, utilizing the minimal number of learning parameters (MNLPs) methodology, the number of NNs’ learning parameters can be significantly reduced. It is rigorously proved that all signals in the closed-loop system are bounded via Lyapunov stability theorem. Finally, simulation and experimental studies are presented to verify the effectiveness and advantages of theoretical results.

The subgame perfect equilibrium in stationary strategies (SSPE) is the most important solution concept in applications of stochastic games, making it imperative to develop efficient methods to compute an SSPE. For this purpose, this paper develops an interior-point differentiable path-following method (IPM), which establishes a connection between an artificial logarithmic barrier game and the stochastic game of interest by adding a homotopy variable. IPM brings several advantages over the existing methods for stochastic games. On the one hand, IPM provides a bridge between differentiable path-following methods and interior-point methods and remedies several issues of an existing homotopy method called the stochastic linear tracing procedure (SLTP). First, the starting stationary strategy profile can be arbitrarily chosen. Second, IPM does not need switching between different systems of equations. Third, the use of a perturbation term makes IPM applicable to all stochastic games rather than generic games only. Moreover, a well-chosen transformation of variables reduces the number of equations and variables by roughly one half. Numerical results show that the proposed method is more than three times as efficient as SLTP. On the other hand, the stochastic game can be reformulated as a mixed complementarity problem and solved by the PATH solver. We employ the proposed IPM and the PATH solver to compute SSPEs. Numerical results evince that for some stochastic games the PATH solver may fail to find an SSPE, whereas IPM is successful in doing so for all stochastic games, which confirms the reliability and stability of the proposed method. Summary of Contribution: This paper incorporates the interior-point methods into a differentiable path-following method for computing stationary equilibria for stochastic games. This novel method brings excellent computational advantages and remedies several issues with the existing methods for stochastic games. We prove the global convergence of the proposed method and employ this method to solve numerous randomly generated stochastic games with different scales. Numerical results further confirm the high efficiency, stability, and universality of this method for stochastic games.

Distance metric learning, which aims at learning an appropriate metric from data automatically, plays a crucial role in the fields of pattern recognition and information retrieval. A tremendous amount of work has been devoted to metric learning in recent years, but much of the work is basically designed for training a linear and global metric with labeled samples. When data are represented with multimodal and high-dimensional features and only limited supervision information is available, these approaches are inevitably confronted with a series of critical problems: 1) naive concatenation of feature vectors can cause the curse of dimensionality in learning metrics and 2) ignorance of utilizing massive unlabeled data may lead to overfitting. To mitigate this deficiency, we develop a semisupervised Laplace-regularized multimodal metric-learning method in this work, which explores a joint formulation of multiple metrics as well as weights for learning appropriate distances: 1) it learns a global optimal distance metric on each feature space and 2) it searches the optimal combination weights of multiple features. Experimental results demonstrate both the effectiveness and efficiency of our method on retrieval and classification tasks.

In this article, a deterministic annealing neural network algorithm is proposed to solve the minimum concave cost transportation problem. Specifically, the algorithm is derived from two neural network models and Lagrange-barrier functions. The Lagrange function is used to handle linear equality constraints, and the barrier function is used to force the solution to move to the global or near-global optimal solution. In both neural network models, two descent directions are constructed, and an iterative procedure for the optimization of the neural network is proposed. As a result, two corresponding Lyapunov functions are naturally obtained from these two descent directions. Furthermore, the proposed neural network models are proved to be completely stable and converge to the stable equilibrium state, therefore, the proposed algorithm converges. At last, the computer simulations on several test problems are made, and the results indicate that the proposed algorithm always generates global or near-global optimal solutions.

In many real-world classification problems, accurate prediction of membership probabilities is critical for further decision making. The probability calibration problem studies how to map scores obtained from one classification algorithm to membership probabilities. The requirement of non-decreasingness for this mapping involves an infinite number of inequality constraints, which makes its estimation computationally intractable. For the sake of this difficulty, existing methods failed to achieve four desiderata of probability calibration: universal flexibility, non-decreasingness, continuousness and computational tractability. This paper proposes a method with shape-restricted polynomial regression, which satisfies all four desiderata. In the method, the calibrating function is approximated with monotone polynomials, and the continuously-constrained requirement of monotonicity is equivalent to some semidefinite constraints. Thus, the calibration problem can be solved with tractable semidefinite programs. This estimator is both strongly and weakly universally consistent under a trivial condition. Experimental results on both artificial and real data sets clearly show that the method can greatly improve calibrating performance in terms of reliability-curve related measures.

Given a group photograph, it is interesting and useful to judge whether the characters in it share specific kinship relation, such as father-daughter, father-son, mother-daughter, or mother-son. Recently, facial image-based kinship verification has attracted wide attention in computer vision. Some metric learning algorithms have been developed for improving kinship verification. However, most of the existing algorithms ignore fusing multiple feature representations and utilizing kernel techniques. In this paper, we develop a novel weighted graph embedding-based metric learning (WGEML) framework for kinship verification. Inspired by the fact that family members usually show high similarity in facial features like eyes, noses, and mouths, despite their diversity, we jointly learn multiple metrics by constructing an intrinsic graph and two penalty graphs to characterize the intraclass compactness and interclass separability for each feature representation, respectively, so that both the consistency and complementarity among multiple features can be fully exploited. Meanwhile, combination weights are determined through a weighted graph embedding framework. Furthermore, we present a kernelized version of WGEML to tackle nonlinear problems. Experimental results demonstrate both the effectiveness and efficiency of our proposed methods.

Clustering ensemble has drawn much attention in recent years due to its ability to generate a high quality and robust partition result. Weighted clustering ensemble and selective clustering ensemble are two general ways to further improve the performance of a clustering ensemble method. Existing weighted clustering ensemble methods assign the same weight to each cluster in a partition of the ensemble. Since the qualities of the clusters in a partition are different, the clusters should be weighted differently. To address this issue, this article proposes a new measure to calculate the similarity between a cluster and a partition. Theoretically, this measure is effective in handling two problems in measuring the quality of a cluster, which are defined as the symmetric problem and the context meaning problem. In addition, some properties of the proposed measure are analyzed. This measure can be easily expanded to a clustering performance measure that calculates the similarity between two partitions. As a result of this measure, we propose a novel selective clustering ensemble framework, which considers the differences between the objective of the ensemble selection stage and the object of the ensemble integration stage in the selective clustering ensemble. To verify the performance of the new measure, we compare the performance of the measure with the two existing measures in weighting clusters. The experiments show that the proposed measure is more effective. To verify the performance of the novel framework, four existing state-of-the-art selective clustering ensemble frameworks are employed as references. The experiments show that the proposed framework is statistically better than the others on 17 UCI benchmark datasets, 8 document datasets, and the Olivetti Face Database.

This paper presents a mixed integer linear programming (MILP) model for the daily planning of home health care (HHC) services provided by the crew of HHC firms. The model takes into account the time window of patients, compatibility requirements, workload limits of staff and distance of patients. The quantitative values of these parameters along with the similarity or difference within the values of a certain parameter are very important to explain the solution of the MILP model. Therefore, we further investigated the relationship between the values of parameters and decisions made in the HHC problem. Our MILP model deals with the scheduling and routing issues along with patient's selection and waiting aspects. The MILP model is solved using the commercial software Ilog-CPLEX and its results are further used to find the association between the relative quantitative gap values linked to certain parameters and decisions obtained through solution of the MILP model. Logistic regression based receiver operating characteristic (ROC) curve approach is used to identify the threshold values for the decision outcome of waiting patients against the related parameters. Subsequent validation and performance evaluation suggest that the proposed thresholds are well defined and can play an important role in making strategies for the HHC system.

Learning from categorical data plays a fundamental role in such areas as pattern recognition, machine learning, data mining, and knowledge discovery. To effectively discover the group structure inherent in a set of categorical objects, many categorical clustering algorithms have been developed in the literature, among which k -modes-type algorithms are very representative because of their good performance. Nevertheless, there is still much room for improving their clustering performance in comparison with the clustering algorithms for the numeric data. This may arise from the fact that the categorical data lack a clear space structure as that of the numeric data. To address this issue, we propose, in this paper, a novel data-representation scheme for the categorical data, which maps a set of categorical objects into a Euclidean space. Based on the data-representation scheme, a general framework for space structure based categorical clustering algorithms (SBC) is designed. This framework together with the applications of two kinds of dissimilarities leads two versions of the SBC-type algorithms. To verify the performance of the SBC-type algorithms, we employ as references four representative algorithms of the k -modes-type algorithms. Experiments show that the proposed SBC-type algorithms significantly outperform the k -modes-type algorithms.

Decision makers often express their evaluations on decision problems with multi-granular linguistic terms. This fact leads to the unification of the multi-granular linguistic terms into a single linguistic set in the literature. However, this unification process increases the complexity of computation and the subjectivity in the determination of transformation functions. To overcome this deficiency, this paper aims to develop a normalized numerical scaling method for determining the semantics of multi-granular linguistic terms in the same domain. We first introduce a class of numerical scaling functions to generate several balanced or unbalanced linguistic sets. Since these scaled linguistic sets have different domains, we then develop a normalized numerical scaling method to form them into the unique interval [0,1]. As a result of this development, two classes of normalized scaling functions are derived from the priori scale information and applications of piecewise linear interpolation and piecewise arc interpolation. Finally, an example is given to illustrate how the method works.

In this paper, we consider lossy mobile ad hoc networks where the data rate of a given flow becomes lower and lower along its routing path. One of the main challenges in lossy mobile ad hoc networks is how to achieve the conflicting goal of increased network utility and reduced power consumption, while without following the instantaneous state of a fading channel. To address this problem, we propose a cross-layer rate-effective network utility maximization (RENUM) framework by taking into account the lossy nature of wireless links and the constraints of rate outage probability and average delay. In the proposed framework, the utility is associated with the effective rate received at the destination node of each flow instead of the injection rate at the source of the flow. We then present a distributed joint transmission rate, link power and average delay control algorithm, in which explicit broadcast message passing is required for power allocation algorithm. Motivated by the desire of power control devoid of message passing, we give a near-optimal power-allocation scheme that makes use of autonomous SINR measurements at each link and enjoys a fast convergence rate. The proposed algorithm is shown through numerical simulations to outperform other network utility maximization algorithms without rate outage probability/average delay constraints, leading to a higher effective rate, lower power consumption and delay. Furthermore, we conduct extensive network-wide simulations in NS-2 simulator to evaluate the performance of the algorithm in terms of throughput, delay, packet delivery ratio and fairness.

A fuzzy granular structure refers to a mathematical structure of the collection of fuzzy information granules granulated from a dataset, while a fuzzy information granularity is used to measure its uncertainty. However, the existing forms of fuzzy information granularity have two limitations. One is that when the fuzzy information granularity of one fuzzy granular structure equals that of the other, one can say that these two fuzzy granular structures possess the same uncertainty, but these two fuzzy granular structures may be not equivalent to each other. The other limitation is that existing axiomatic approaches to fuzzy information granularity are still not complete, under which when the partial order relation among fuzzy granular structures cannot be found, their coarseness/fineness relationships will not be revealed. To address these issues, a so-called fuzzy granular structure distance is proposed in this study, which can well discriminate the difference between any two fuzzy granular structures. Besides this advantage, the fuzzy granular structure distance has another important benefit: It can be used to establish a generalized axiomatic constraint for fuzzy information granularity. By using the axiomatic constraint, the coarseness/fineness of any two fuzzy granular structures can be distinguished. In addition, through taking the fuzzy granular structure distances of a fuzzy granular structure to the finest one and the coarsest one into account, we also can build a bridge between fuzzy information granularity and fuzzy information entropy. The applicable analysis on 12 real-world datasets shows that the fuzzy granular structure distance and the generalized fuzzy information granularity have much better performance than existing methods.

Novelty detection models aim to find the minimum volume set covering a given probability mass. This paper proposes a robust single-class support vector machine (SSVM) for novelty detection, which is mainly based on the worst case conditional value-at-risk minimization. By assuming that every input is subject to an uncertainty with a specified symmetric support, this robust formulation results in a maximization term that is similar to the regularization term in the classical SSVM. When the uncertainty set is l1 -norm, l∞ -norm or box, its training can be reformulated to a linear program; while the uncertainty set is l2 -norm or ellipsoidal, its training is a tractable second-order cone program. The proposed method has a nice consistent statistical property. As the training size goes to infinity, the estimated normal region converges to the true provided that the magnitude of the uncertainty set decreases in a systematic way. The experimental results on three data sets clearly demonstrate its superiority over three benchmark models.

Many real data increase dynamically in size. This phenomenon occurs in several fields including economics, population studies, and medical research. As an effective and efficient mechanism to deal with such data, incremental technique has been proposed in the literature and attracted much attention, which stimulates the result in this paper. When a group of objects are added to a decision table, we first introduce incremental mechanisms for three representative information entropies and then develop a group incremental rough feature selection algorithm based on information entropy. When multiple objects are added to a decision table, the algorithm aims to find the new feature subset in a much shorter time. Experiments have been carried out on eight UCI data sets and the experimental results show that the algorithm is effective and efficient.

As a leading partitional clustering technique, k-modes is one of the most computationally efficient clustering methods for categorical data. In the k-modes, a cluster is represented by a "mode," which is composed of the attribute value that occurs most frequently in each attribute domain of the cluster, whereas, in real applications, using only one attribute value in each attribute to represent a cluster may not be adequate as it could in turn affect the accuracy of data analysis. To get rid of this deficiency, several modified clustering algorithms were developed by assigning appropriate weights to several attribute values in each attribute. Although these modified algorithms are quite effective, their convergence proofs are lacking. In this paper, we analyze their convergence property and prove that they cannot guarantee to converge under their optimization frameworks unless they degrade to the original k-modes type algorithms. Furthermore, we propose two different modified algorithms with weighted cluster prototypes to overcome the shortcomings of these existing algorithms. We rigorously derive updating formulas for the proposed algorithms and prove the convergence of the proposed algorithms. The experimental studies show that the proposed algorithms are effective and efficient for large categorical datasets.

This paper presents a decision-making model described by a recurrent neural network for dynamic portfolio optimization. The portfolio-optimization problem is first converted into a constrained fractional programming problem. Since the objective function in the programming problem is not convex, the traditional optimization techniques are no longer applicable for solving this problem. Fortunately, the objective function in the fractional programming is pseudoconvex on the feasible region. It leads to a one-layer recurrent neural network modeled by means of a discontinuous dynamic system. To ensure the optimal solutions for portfolio optimization, the convergence of the proposed neural network is analyzed and proved. In fact, the neural network guarantees to get the optimal solutions for portfolio-investment advice if some mild conditions are satisfied. A numerical example with simulation results substantiates the effectiveness and illustrates the characteristics of the proposed neural network.

Abstract The notion of information systems provides a convenient tool for knowledge representation of objects in terms of their attribute values, while partial ordering is usually used to research the rough monotonicity of an uncertainty measure in information systems. In this paper, we first reveal the limitations of existing partial orderings to describe information granulations in information systems with several illustrative examples. Then, a generalized partial ordering with a set‐size nature is proposed to overcome their shortcoming and some of its important properties are derived. Finally, we prove that several existing information granulations all satisfy the granulation monotonicity induced by the proposed partial ordering. The presented partial ordering appears to be well suited to characterize the nature of information granulations in an information system. These results will be very helpful for studying granular computing and uncertainty in information systems.

$K$-means is a partitional clustering technique that is well-known and widely used for its low computational cost. The representative algorithms include the hard$k$-means and the fuzzy$k$-means. However, the performance of these algorithms tends to be affected by skewed data distributions, i.e., imbalanced data. They often produce clusters of relatively uniform sizes, even if input data have varied cluster sizes, which is called the “uniform effect.” In this paper, we analyze the causes of this effect and illustrate that it probably occurs more in the fuzzy$k$-means clustering process than the hard$k$-means clustering process. As the fuzzy index$m$increases, the “uniform effect” becomes evident. To prevent the effect of the “uniform effect,” we propose a multicenter clustering algorithm in which multicenters are used to represent each cluster, instead of one single center. The proposed algorithm consists of the three subalgorithms: the fast global fuzzy$k$-means, Best M-Plot, and grouping multicenter algorithms. They will be, respectively, used to address the three important problems: 1) How are the reliable cluster centers from a dataset obtained? 2) How are the number of clusters which these obtained cluster centers represent determined? 3) How is it judged as to which cluster centers represent the same clusters? The experimental studies on both synthetic and real datasets illustrate the effectiveness of the proposed clustering algorithm in clustering balanced and imbalanced data.

This paper is concerned with the problem of delay-dependent H∞filter design for a class of discrete-time nonlinear interconnected system with time-varying delays via the Takagi-Sugeno (T-S) fuzzy model. The T-S fuzzy model consists ofNtime-delay T-S fuzzy subsystems, and a decentralized H∞filter is designed for each subsystem. Based on the delay-dependent piecewise Lyapunov-Krasovskii functional (DDPLKF) and with an improved free-weighting matrix technique, the delay-dependent stability and a prescribed H∞performance index are guaranteed for the overall filtering error system. A sufficient condition for the existence of such a filter is established by using linear matrix inequalities (LMIs) that are numerically feasible. Two numerical examples are given to demonstrate the effectiveness and advantage of the proposed approach.

In this paper we introduce a new rule for butterfly subdivision scheme which overcome a shortcoming with the sharp features. For butterfly subdivision scheme, many features will be lost after several times of refinement. This new rule solved this problem by removing the position of new interpolated vertices. And our method can also implement in some other subdivision schemes.

When the leader's objective function of a nonlinear bilevel programming problem is nondifferentiable and the follower's problem of it is nonconvex, the existing algorithms cannot solve the problem. In this paper, a new effective evolutionary algorithm is proposed for this class of nonlinear bilevel programming problems. First, based on the leader's objective function, a new fitness function is proposed that can be easily used to evaluate the quality of different types of potential solutions. Then, based on Latin squares, an efficient crossover operator is constructed that has the ability of local search. Furthermore, a new mutation operator is designed by using some good search directions so that the offspring can approach a global optimal solution quickly. To solve the follower's problem efficiently, we apply some efficient deterministic optimization algorithms in the MATLAB Toolbox to search for its solutions. The asymptotically global convergence of the algorithm is proved. Numerical experiments on 25 test problems show that the proposed algorithm has a better performance than the compared algorithms on most of the test problems and is effective and efficient.

Zadeh’s seminal work in theory of fuzzy-information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it. This has led to an interesting research topic: granular computing (GrC). Although many excellent research contributions have been made, there remains an important issue to be addressed: What is the essence of measuring a fuzzy-information granularity of a fuzzy-granular structure? What is needed to answer this question is an axiomatic constraint with a partial-order relation that is defined in terms of the size of each fuzzy-information granule from a fuzzy-binary granular structure. This viewpoint is demonstrated for fuzzy-binary granular structure, which is called the binary GrC model by Lin. We study this viewpoint from from five aspects in this study, which are fuzzy BINARY-granular-structure operators, partial-order relations, measures for fuzzy-information granularity, an axiomatic approach to fuzzy-information granularity, and fuzzy-information entropies.

A fundamental assumption often made in unsupervised learning is that the problem is static, i.e., the description of the classes does not change with time. However, many practical clustering tasks involve changing environments. It is hence recognized that the methods and techniques to analyze the evolving trends for changing environments are of increasing interest and importance. Although the problem of clustering numerical time-evolving data is well-explored, the problem of clustering categorical time-evolving data remains as a challenging issue. In this paper, we propose a generalized clustering framework for categorical time-evolving data, which is composed of three algorithms: a drifting-concept detecting algorithm that detects the difference between the current sliding window and the last sliding window, a data-labeling algorithm that decides the most-appropriate cluster label for each object of the current sliding window based on the clustering results of the last sliding window, and a cluster-relationship-analysis algorithm that analyzes the relationship between clustering results at different time stamps. The time-complexity analysis indicates that these proposed algorithms are effective for large datasets. Experiments on a real dataset show that the proposed framework not only accurately detects the drifting concepts but also attains clustering results of better quality. Furthermore, compared with the other framework, the proposed one needs fewer parameters, which is favorable for specific applications.

In this paper, based on a one-neuron recurrent neural network, a novel k-winners-take-all ( k -WTA) network is proposed. Finite time convergence of the proposed neural network is proved using the Lyapunov method. The k-WTA operation is first converted equivalently into a linear programming problem. Then, a one-neuron recurrent neural network is proposed to get the kth or (k+1)th largest inputs of the k-WTA problem. Furthermore, a k-WTA network is designed based on the proposed neural network to perform the k-WTA operation. Compared with the existing k-WTA networks, the proposed network has simple structure and finite time convergence. In addition, simulation results on numerical examples show the effectiveness and performance of the proposed k-WTA network.

The original rough-set model is primarily concerned with the approximations of sets described by a single equivalence relation on a given universe. With granular computing point of view, the classical rough-set theory is based on a single granulation. This correspondence paper first extends the rough-set model based on a tolerance relation to an incomplete rough-set model based on multigranulations, where set approximations are defined through using multiple tolerance relations on the universe. Then, several elementary measures are proposed for this rough-set framework, and a concept of approximation reduct is introduced to characterize the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in this rough-set model. Finally, several key algorithms are designed for finding an approximation reduct.

In general, due to the interaction among subsystems, it is difficult to design an Hinfinity filter for nonlinear interconnected systems. This paper introduces a decentralized Hinfinity fuzzy filter design for nonlinear interconnected systems with multiple time delays via T-S fuzzy models. The T-S fuzzy model consists of N time-delay T-S fuzzy subsystems. The decentralized Hinfinity filter is designed based on this model, which the asymptotic stability and a prescribed Hinfinity performance index are guaranteed for the overall filtering error system. A sufficient condition for the existence of such a filter is established by using linear matrix inequalities that are numerically feasible. A simulation example is given to show the effectiveness of this approach.

Designing efficient algorithms for difficult multi-objective optimization problems is a very challenging problem. In this paper a new clustering multi-objective evolutionary algorithm based on orthogonal and uniform design is proposed. First, the orthogonal design is used to generate initial population of points that are scattered uniformly over the feasible solution space, so that the algorithm can evenly scan the feasible solution space once to locate good points for further exploration in subsequent iterations. Second, to explore the search space efficiently and get uniformly distributed and widely spread solutions in objective space, a new crossover operator is designed. Its exploration focus is mainly put on the sparse part and the boundary part of the obtained non-dominated solutions in objective space. Third, to get desired number of well distributed solutions in objective space, a new clustering method is proposed to select the non-dominated solutions. Finally, experiments on thirteen very difficult benchmark problems were made, and the results indicate the proposed algorithm is efficient.

An arbitrary starting homotopy-like simplicial algorithm is developed for computing an integer point in a polytope given by $P=\{x\mid Ax\leq b\}$ satisfying that each row of A has at most one positive entry. The algorithm is derived from an introduction of an integer labeling rule and an application of a triangulation of the space $R^n\times[0,1]$. It consists of two phases, one of which forms an $(n+1)$-dimensional pivoting procedure and the other an n-dimensional pivoting procedure. Starting from an arbitrary integer point in $R^n\times\{0\}$, the algorithm interchanges from one phase to the other, if necessary, and follows a finite simplicial path that either leads to an integer point in the polytope or proves that no such point exists.

This paper describes a decentralizedHinfinfilter design for discrete-time interconnected fuzzy systems based on piecewise-quadratic Lyapunov functions. The systems consist ofJdiscrete-time interconnected Takagi-Sugeno (T-S) fuzzy subsystems, and a decentralizedHinfinfilter is designed for each subsystem. It is shown that the stability of the overall filtering-error system withHinfinperformance can be established if a piecewise-quadratic Lyapunov function can be constructed. Moreover, the parameters of filters can be obtained by solving a set of linear matrix inequalities that are numerically feasible. Two simulation examples are given to show the effectiveness of the proposed approach.

Rough set theory is a relatively new mathematical tool for computer applications in circumstances characterized by vagueness and uncertainty. In this paper, we address uncertainty of rough sets for incomplete information systems. An axiom definition of knowledge granulation for incomplete information systems is obtained, under which a measure of uncertainty of a rough set is proposed. This measure has some nice properties such as equivalence, maximum and minimum. Furthermore, we prove that the uncertainty measure is effective and suitable for measuring roughness and accuracy of rough sets for incomplete information systems.

This paper considers the robustHinfincontrol of uncertain discrete-time fuzzy systems with time delays based on piecewise Lyapunov--Krasovskii functionals. It is shown that the stability withHinfindisturbance attenuation performance can be established for the closed-loop fuzzy control systems if there exists a piecewise Lyapunov--Krasovskii functional, and moreover, the functional and the corresponding controller can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A numerical example is given to demonstrate the efficiency and the advantage of the proposed method.

In this paper, the level-set evolution is exploited in the design of a novel evolutionary algorithm (EA) for global optimization. An application of Latin squares leads to a new and effective crossover operator. This crossover operator can generate a set of uniformly scattered offspring around their parents, has the ability to search locally, and can explore the search space efficiently. To compute a globally optimal solution, the level set of the objective function is successively evolved by crossover and mutation operators so that it gradually approaches the globally optimal solution set. As a result, the level set can be efficiently improved. Based on these skills, a new EA is developed to solve a global optimization problem by successively evolving the level set of the objective function such that it becomes smaller and smaller until all of its points are optimal solutions. Furthermore, we can prove that the proposed algorithm converges to a global optimizer with probability one. Numerical simulations are conducted for 20 standard test functions. The performance of the proposed algorithm is compared with that of eight EAs that have been published recently and the Monte Carlo implementation of the mean-value-level-set method. The results indicate that the proposed algorithm is effective and efficient.

A Lagrange multiplier and Hopfield-type barrier function method is proposed for approximating a solution of the traveling salesman problem. The method is derived from applications of Lagrange multipliers and a Hopfield-type barrier function and attempts to produce a solution of high quality by generating a minimum point of a barrier problem for a sequence of descending values of the barrier parameter. For any given value of the barrier parameter, the method searches for a minimum point of the barrier problem in a feasible descent direction, which has a desired property that lower and upper bounds on variables are always satisfied automatically if the step length is a number between zero and one. At each iteration, the feasible descent direction is found by updating Lagrange multipliers with a globally convergent iterative procedure. For any given value of the barrier parameter, the method converges to a stationary point of the barrier problem without any condition on the objective function. Theoretical and numerical results show that the method seems more effective and efficient than the softassign algorithm.