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The essence of human is social relation. In computer, relation can be represented by graph. It may be possible to divide a group of individuals into a few of communities according to the intensity of relation among individuals. The relation between individuals in the same community should be strong, while that between individuals in different communities should be weak. That the intensity of relation between two persons is high means that they trust each other; therefore, strong relation can be used to commit crime, for instance, corruption. Consequently, finding out strong relation can help auditor to reduce difficulty of assuring them of reliability of auditing. Seeing that those algorithms that involve iteration process have fatal defect, the authors of this paper introduce a method used to divide a group into a few of communities based on spectral clustering. This method has the advantage of high speed which means that it has favorable performance when it is used to cope with tremendous amount of data.

An iteration-free fractal mating coding (IFMC) method for mutual image compression is proposed in this paper. To avoid the iteration process in the decoder, the mean image is used to generate the domain pool in both the encoder and decoder. The proposed IFMC method successfully solves the problems of long iteration duration and large memory requirement in the decoding stage in our previous fractal mating coding (FMC) method. Computer simulation results show the proposed IFMC method outperforms the previous FMC method in both the coding performance and decoding speed.

In this paper, we study the convergence and Δ —convergence results of K —iteration process for Lipschizian self-mapping with L ≥ 1in CAT (H) spaces, H > 0.

This essay synthesizes more than a decade of research, most of which has been published, on the teaching and learning of floating and sinking (FS) phenomena. The research is comprised of the iterative design, development, implementation and evaluation of a Teaching-Learning sequence (TLS) for the teaching and learning of density within FS phenomena. It was initiated within the frame of the European Community supported “Materials Science” project. Due to the many, different aspects of the project, each publication has focused on a particular part of the study (e.g., effectiveness and the iteration process). The didactic transformation for the teaching of FS phenomena is presented and discussed here. In doing so, it is essential to mention: (a) the students’ ideas as the main cause of the scientific knowledge transformation, (b) the scientific/reference knowledge, and (c) the knowledge to be taught and its limitations. Thus, we intend to describe and justify the didactic transformation process and briefly synthesize the published (from previous papers) and unpublished results to show its effectiveness.

In this paper, we introduce a new modified iteration process for approximating common endpoint of a multivalued α-nonexpansive mapping and a multivalued mapping satisfying condition (E′) in uniformly convex hyperbolic spaces. As a by-product, we improve the main results of Panyanak (2018) with a new and faster algorithm for two multivalued mappings. Moreover, we give a numerical example to substantiate our results. Our work is new and holds simultaneously in uniformly convex Banach spaces as well as CAT(0) spaces.

The convergence speed of analytic distributed multi-objective optimization algorithms should be higher when solving distributed multi-objective optimization algorithms. An adaptive operation is introduced into the only analytic distributed multi-objective optimization algorithm, which is an interchange objective value method. Therefore, an adaptive interchange objective value method is proposed for distributed multi-objective optimization problems. The proposed adaptive interchange objective value method updates the reward coefficients of a basic analytical distributed multi-objective optimization algorithm in the iteration process of solving distributed multi-objective optimization problems. The adaptive interchange objective value method obtains multiple satisfy optimal objectives for multiple subsidiary distributed multi-objective optimization problems security and quickly. To verify the feasibility and effectiveness of the adaptive interchange objective value method for the analytical distributed multi-objective optimization problems, the analytical distributed multi-objective optimal power flow problems under IEEE 118-bus, IEEE 300-bus power system and the medium part of the European system with 1472-bus test system are simulated. The numerical simulation results under these three cases show that the proposed adaptive interchange objective value method can obtain multiple distributed objectives for analytical distributed multi-objective optimal power flow problems security and quickly.

In this paper, we focus on the periodic points during the iteration process to understand the convergence characteristics of the Mandelbrot set. A new internal structure of the Mandelbrot set is obtained. In addition, the influences of the number of initial iteration points and computational accuracy on this structure are investigated. The results show that the new internal structure is stable no matter how the computational accuracy and the number of initial iteration points change. However, the boundaries of subsets of the Mandelbrot set are sensitive to the computational accuracy. This finding reveals the true convergence structure of the Mandelbrot set during numerical simulations, which differs from the theoretical convergence point distribution. Thus, it helps us further understand the convergence mechanism of the Mandelbrot set.

An approximate method is presented, using (with suitable corrections) only the compliance’s poles that are located nearly positive real axis in the complex frequency plane, for solving transient vibration problems containing different damping mechanisms. For determination of poles and its residues, a new method based on consideration of known values of a function in the vicinity of a pole is suggested which allows achieving a required accuracy due to an iteration process. Simple and sufficiently accurate approximate formulas are obtained for treatment mechanical systems with different damping mechanisms under the action of impulse loads. A particular attention is paid to the model, which can be considered as a correction of the ideal hysteretic model.

In the present article, we establish a semilocal convergence theorem for the S-iteration process of Newton---Kantorovich like in Banach space setting for solving nonlinear operator equations and discuss its semilocal convergence analysis. We apply our result to solve the Fredholm-integral equations.

We propose a new modified proximal point algorithm combined with Halpern's iteration process for nonexpansive mappings in the framework of CAT0 spaces. We establish a strong convergence theorem under some mild conditions. Our results extend some known results which appeared in the literature.

Electric energy storage is one of the most promising and convenient solutions for energy cost management of buildings by shifting electric loads. This paper investigates electric energy storage (EES) size planning to minimize the total electricity cost of buildings, including investment. This study discusses EES planning considering an EES operation, especially focusing on an economically optimized decision of EES capacity. We first formulate a general electricity cost minimization problem with the electricity bill of buildings and EES cost, and decompose the problem into two parts for EES planning and EES operations. The EES planning algorithm based on the iteration process is proposed considering an ESS operation. The simulation demonstrates that the proposed EES planning can perform close to the optimal exhaustive search algorithm.

In this paper, we prove a weak convergence theorem of the implicit iteration process for the semigroups of Lipschitz pseudocontractive mappings in uniformly convex Banach spaces with the Opial property.

The iteration algorithm is used to solve systems of linear algebraic equations by the Monte-Carlo method. Each next iteration is simulated as a random vector such that its expectation coincides with the Seidel approximation of the iteration process. We deduce a system of linear equations such that mutual correlations of components of the limit vector and correlations of two iterations satisfy them. We prove that limit dispersions of the random vector of solutions of the system exist and are finite.

Alternating-projection-type dual-space algorithms have a clear construction, but are susceptible to stagnation and, thus, inefficient for solving the phase problemab initio. To improve this behaviour new omit maps are introduced, which are real-space perturbations applied periodically during the iteration process. The omit maps are calledvolumic, because they delete some predetermined subvolume of the unit cell without searching for atomic regions or analysing the electron density in any other way. The basic algorithms of positivity, histogram matching and low-density elimination are tested by their solution statistics. It is concluded that, while all these algorithms based on weak constraints are practically useless in their pure forms, appropriate volumic omit maps can transform them to practically useful methods. In addition, the efficiency of the already useful reflector-type charge-flipping algorithm can be further improved. It is important that these results are obtained by using non-sharpened structure factors and without any weighting scheme or reciprocal-space perturbation. The mathematical background of volumic omit maps and their expected applications are also discussed.

In this paper, the magnetic field design of a superconducting fixed-frequency isochronous cyclotron with four spiral sectors for proton therapy was performed. The basic parameters of the isochronous cyclotron magnet and its characteristics are described. Several magnet sectors were designed during the iteration process to achieve an isochronous magnetic field. The result of 3-D magnetic field simulation using OPERA-3D and the analysis of the designed magnetic field are presented. A special geometry was introduced into the sector gap to achieve the isochronous condition, and the spiral angle was adjusted to control the vertical focusing parameter. The equilibrium orbits were checked for the designed magnetic field, and a single particle tracking simulation was performed with CYCLONE code.

Consider the solutions of the defocusing cubic wave equation ∂ t t u − Δ u = − u 3 , with real data in H s ( T 2 ) × H s − 1 ( T 2 ) , s > 2 5 . We prove that the solutions exist globally in time by contradiction. Assuming that one of the maximal times of existence is finite, we prove that the Sobolev norm of each of these solutions is bounded in an open neighborhood of it by using the I-method (see e.g. [4] ). It consists of introducing a mollified energy and estimating its growth in this neighborhood. The growth is first estimated on small subintervals by means of dispersive bounds; then it is estimated on the whole interval by an iteration process. In order to enlarge the size of subintervals on which the dispersive bounds hold, we interpolate with the potential bound for the low frequency part. Our results complement those obtained in [1] .

In this paper, we introduce a new iteration process for solving fixed point problem of Bregman strongly nonexpansive mappings, and then we study a strong convergence theorem for a common fixed point of Bregman strongly nonexpansive mappings in the framework of reflexive Banach spaces.

Simultaneous adjustment of all cutter locations in a tool path using meta-heuristic algorithms provides a systematic approach to reducing machining errors in 5-axis flank machining of ruled surfaces. However, these algorithms experienced unsatisfactory quality of optimal solutions and lengthy search time in high-dimensional search space. To solve these problems, we propose an iterative optimization scheme that progressively simplifies solution space by sampling techniques. Akaike information criterion (AIC) is used to screen significant factors from sampling data. An electromagnetism-like mechanism (EM) algorithm searches through a simplified solution space constructed only using these factors. An iteration process consisting of such sampling, screening, and searching steps repeats several times until final optimal solutions are obtained. Test results of representative surfaces validate the effectiveness of the proposed scheme. Both solution quality and search efficiency are improved comparing to those produced by previous studies. This work enhances the practicality of optimization-driven tool path planning in 5-axis flank machining.

In this paper, we study traveling waves connecting the infection-free equilibrium state and the endemic equilibrium state for a diffusive SIR model with delay and saturated incidence rate. Since this system does not enjoy the comparison principle, we will use an iteration process to construct a pair of upper and lower solutions. With the aid of the pair of upper and lower solutions, we can apply the Schauder fixed point theorem to construct a family of solutions of the truncated problems, which, via the limiting argument, can generate the traveling wave. Indeed, we show that there exists c ⁎ > 0 such that this system admits a traveling wave solution with speed c iff c ≥ c ⁎ .

In this paper we prove the strong and Delta-convergence theorems of an iteration process of Khan et al. (J. Appl. Math. Comput. 35 (2011) 607-616) for three finite families of total asymptotically nonexpansive nonself mappings in a hyperbolic space. Moreover we obtain the data dependence result of this iteration for contractive-like mappings under some suitable conditions. Also we present some examples to support the results proved herein. Our results extend and improve some recent results announced in the current literature. https://doi.org/10.2298/FIL1603569S

Integration of centralized wind farms may induce cascading tripping incidents, which brings about critical challenges to secure operation of power grid. One of the major risks is the lack of coordinated voltage/reactive power control for multiple wind farms. Therefore, a robust master-slave two-level coordinated static voltage security region (VSR) is proposed in this paper, along with detailed modeling of the network topology and wind farms to achieve coordination among wind farms. The slave VSR on the wind-farm-side is designed to guarantee the operation of each wind unit within a secure range in order to protect wind units from serious over-voltage and low-voltage problems, whereas the master system-wide VSR aims to coordinate centralized wind farms to ensure security both under normal operating conditions and $\boldsymbol {N - 1}$ contingency conditions. Furthermore, a hierarchically iterative coordination method is introduced to compute the voltage security range of each wind farm through the iteration process. A modified six-bus system with two wind farms, a real system from Northern China, and a large system with 972 wind units are studied, and results verify the effectiveness of the proposed two-level VSR and the hierarchically iterative coordination method.

This paper presents a new numerical method to optimize the mix proportion of self-compacting concrete (SCC) at paste level based on a paste rheological threshold theory that the SCC formulated paste must satisfy the criteria to ensure SCC with good flowability and segregation resistance. A total of 12 mini-slump flow tests were initially conducted, and relevant rheological characteristics were derived. The correlation between mixture variables and rheological characteristics was determined using a bilinear interpolation method. A self-compacting area at paste level was defined by choosing the mixture whose rheological characteristics satisfied the criteria. The economical mixture was obtained from the area, and an iteration process was introduced to ensure that the rheological characteristics of this mixture satisfied the criteria. Finally, an optimized SCC mixture from an economical aspect was achieved, and the robustness of the mixture was also analyzed.

In this Letter, we propose a novel configuration and design method of freeform, dual fields-of-view (FOVs), dual focal lengths, off-axis three-mirror zoom imaging systems. The switch of the two zooms is achieved by rotating a single mirror element. The design of a freeform, dual focal lengths zoom system is realized by a point-by-point design approach for the first time to our knowledge. This method enables the direct design of freeform surfaces from initial planes using given system specifications and configuration, and the designed system can be taken as a good starting point for further optimization. A freeform, dual FOVs, dual focal lengths, off-axis three-mirror zoom system is demonstrated. The F-numbers of the two zooms are 2 and 2.4. The dual FOVs are 3°×3° and 2.5°×2.5°. After final optimization, both of the zooms achieve high performances.

In this paper, we introduce a new three-step iteration scheme and establish convergence results for approximation of fixed points of nonexpansive mappings in the framework of Banach space. Further, we show that the new iteration process is faster than a number of existing iteration processes. To support the claim, we consider a numerical example and approximated the fixed point numerically by computer using MATLAB.

Mann&rsquo, s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fr&eacute, chet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann&rsquo, s iteration process and its variants.

We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors. The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case. A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.

In this research, we introduce new iteration process for different types of mappings and introduce a concept of expansion mapping, it is independent of . Also, we study the convergences for these iterations to common fixed point in real Hilbert spaces.

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by x n + 1 = t n T ϕ ( n ) ( x n ) + ( 1 − t n ) x n , for n ∈ N , when T is a monotone asymptotically nonexpansive self-mapping.

How can design thinking be taught in a Massive Open Online Course (MOOC)? The research team took an iterative approach at designing an online course about the design research phase and built a prototype MOOC for testing. We applied three measurement tools (Course Evaluation Survey, Skill Confidence Rating, Qualitative Interviews) and categorized the collected feedback. This process resulted in 57 iteration tasks, which were implemented in the public version of the MOOC. From over 5000 participants, 3034 learners participated actively in the public MOOC. 84.75% of survey participants were satisfied with the MOOC. In this chapter, we describe the iteration process and first results from the public MOOC.

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by x n + 1 = t n T ϕ ( n ) ( x n ) + ( 1 − t n ) x n , for n ∈ N , when T is a monotone asymptotically nonexpansive self-mapping.

In this paper, we prove some weak and strong convergence results for generalized [Formula: see text]-nonexpansive mappings using [Formula: see text] iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196].

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates [Formula: see text] connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when [Formula: see text] is even or odd. Finally, we obtain the analytical expression of the eigentime identity and the scalings with network size of the weighted scale-free networks.

The satellite layout optimization design (SLOD) problem is a kind of three-dimensional layout problems with complex performance constraints and known as a NP-hard problem. To solve SLOD problems efficiently and effectively, two types of hybrid optimization algorithm based on differential evolution (DE) are proposed in this paper. Concerning the design requirements of satellite attitude control subsystem, the SLOD problem is formulated, aiming to improve the overall mass characteristics of satellite. To explore the layout design space globally, the DE algorithm is utilized as the main framework of the proposed hybrid algorithm. Then in order to improve the local exploitation capability and algorithm robustness, sequential quadratic programming (SQP), as a gradient-based method, is combined with DE in two unique ways, comprising two types of hybrid algorithm. In the first type of hybrid algorithm (denoted by DESQP), SQP is performed when iteration process of DE has finished and only the final solution of DE is used as the initial point of SQP, the purpose of which is to locate the most promising area of optimum with DE first and then make a rapid exploitation around the quasi-optimum. In the second type of hybrid algorithm (denoted by DESQPDE), SQP is performed in the specific iteration of DE and all the current-generation population individuals are used as the initial points, the purpose of which is to accelerate the evolution process while holding the diversity of the population and to enhance the robustness. Finally, the efficacy and robustness of the proposed hybrid algorithms are compared with classical DE and also validated by two three-dimensional satellite layout cases with 14 and 40 components, respectively.

In this paper, some strong and Δ-convergence results are proved for Suzuki generalized nonexpansive mappings in the setting of $\mathit{CAT}(0)$ spaces using the K iteration process. We also give an example to show the efficiency of the K iteration process. Our results are the extension, improvement and generalization of many well-known results in the literature of fixed point theory in $\mathit{CAT}(0)$ spaces.

This short appendix summarizes some of the properties used in this work as well as measurements of permittivity performed for the purpose of ascertaining published values. In some cases, the permittivity was calculated from simulations. Since the resonant frequency of the actual material was measured and given the dimensions of the medium, a simple process of iteration on the permittivity in the simulation until the simulated resonant frequency matched the measured resonant frequency provided the correct value for the material. In other cases, simulations were performed to ascertain the exact value of permittivity.

The paper deals with the use of Projective Adaptive Resonance Theory neural network (PART) to generate dictionary of context-dependent keyword pairs from text document. The principle of automatic generation of context-dependent keywords pairs is based on a modified algorithm of clustering MA-PART, which was proposed by the authors not long ago to generate the dictionary of single keywords. The proposed model allows clustering in a single iteration process of two words in the input vector. Context dependency of test pair of words is determined on the basis of determination of pair membership within the same cluster.

Thermal performance of five bicycle helmets was evaluated with a thermal manikin head with six zones. Evaluation was made with physical and virtual experimental methods. Ambient temperature maintained at 24 degrees C and surface temperature of the thermal manikin head was set to 34 degrees C. Experiments were performed for air velocities of 1.6 m/s and 6 m/s. Heat transfer (W) of four thermal zones was recorded for five helmets and compared with a nude thermal manikin head to assess thermal performance. Virtual experiments were performed using commercial CFD codes with a realizable k-epsilon turbulence model. Correlation coefficients of 0.78 (1.6 m/s) and 0.79 (6 m/s) were found between physical and virtual experiments. A combined physical and virtual evaluation methodology allows creating a design iteration process with virtual prototypes, physical prototypes and commercially available helmets.

In this paper, we modified multi-step procedure to find approximation fixed point of pairwise generalized nonexpansive mappings in CAT(0) spaces. We also prove both strong and \(\varDelta \)-convergence theorems for such a mapping with under mild conditions.

In this paper, we propose a faster iteration process applied to generalized support vector machine. First, we introduce the theory of support vector machine and give the definition of generalized support vector machine, which is equivalent to the variational inequality problem and fixed point problem. Also, we introduce an iterative process for finding the set of solutions of variational inequality problem. The result show that the iterative sequence converges strongly to the solution of the fixed point problem and the classical variational inequality problem. Obviously, the solution solve the generalized support vector machine.

The global automotive industry is facing challenges in several key areas, including energy, emissions, safety, and affordability. Lightweighting is one of the key strategies used to address these challenges. Maximizing the weight reduction (i.e., minimizing vehicle weight) requires a systems-engineering design optimization and iteration process that combines material properties and manufacturing processes to meet product requirements at the lowest mass and/or cost. Advanced high-strength steels, aluminum and magnesium alloys, and carbon-fiber-reinforced polymers have emerged as important materials for automotive lightweighting. This article presents examples of how coupling materials science with innovative manufacturing processes can provide lightweight solutions in automotive engineering.

The relaxation factor is a key parameter in gradient-based inversion and optimization methods, as well as in solving nonlinear equations using iterative techniques. In gradient-based inversion methods, the relaxation factor directly affects the inversion efficiency and the convergence stability. In general, the bigger the relaxation factor is, the faster the inversion process is. However, divergences may occur if the relaxation factor is too big. Therefore, there should be an optimal value of the relaxation factor at each iteration, guaranteeing a high inversion efficiency and a good convergence stability. In the present work, an optimization technique is proposed, using which the relaxation factor is adaptively updated at each iteration, rather than a constant during the whole iteration process. Based on this, a new inverse analysis method is developed for solving multi-dimensional transient nonlinear inverse heat conduction problems. One- and two-dimensional transient nonlinear inverse heat conduction problems are involved, and the instability issues occurred in the previous works are reconsidered. The results show that the new inverse analysis method in the present work has the same high accuracy, the same good robustness, and a higher inversion efficiency, compared with the previous least-squares method. Most importantly, the new method is more stable by innovatively optimizing and adaptively updating the relaxation factor at each iteration.

Based on the Hermitian and skew-Hermitian splitting (HSS), we come up with a generalized HSS iteration method with a flexible shift-parameter for solving the non-Hermitian positive definite system of linear equations. This iteration method utilizes the optimization technique to obtain the optimal value of the flexible shift-parameter at iteration process. Both theory and experiment have shown that the new strategy is efficient.

Purpose – In multidisciplinary design optimization (MDO), if the relationships between design variables and some output parameters, which are important performance constraints, are complex implicit problems, plenty of time should be spent on computationally expensive simulations to identify whether the implicit constraints are satisfied with the given design variables during the optimization iteration process. The purpose of this paper is to propose an ensemble of surrogates-based analytical target cascading (ESATC) method to tackle such MDO engineering design problems with reduced computational cost and high optimization accuracy. Design/methodology/approach – Different surrogate models are constructed based on the sample point sets obtained by Latin hypercube sampling (LHS) method. Then, according to the error metric of each surrogate model, the repeated ensemble of surrogates is constructed to approximate the implicit objective functions and constraints. Under the framework of analytical target cascading (ATC), the MDO problem is decomposed into several optimization subproblems and the function of analysis module of each subproblem is simulated by repeated ensemble of surrogates, working together to find the optimum solution. Findings – The proposed method shows better modeling accuracy and robustness than other individual surrogate model-based ATC method. A numerical benchmark problem and an industrial case study of the structural design of a super heavy vertical lathe machine tool are utilized to demonstrate the accuracy and efficiency of the proposed method. Originality/value – This paper integrates a repeated ensemble method with ATC strategy to construct the ESATC framework which is an effective method to solve MDO problems with implicit constraints and black-box objectives.

In this paper, we study three-step iteration process for Ciric-quasi contractive operator and establish strong convergence theorems for above mentioned operator and schemes in the setting of CAT(0) spaces. Our result extends and generalizes some previous work from the existing literature (see, e.g., [4, 30] and some others).

On the premise of appropriate learning accuracy, the number of the neurons of a neural network should be as less as possible (constructional sparsification), so as to reduce the cost, and to improve the robustness and the generalization accuracy. We study the constructional sparsification of feedforward neural networks by using regularization methods. Apart from the traditional L 1 regularization for sparsification, we mainly use the L 1/2 regularization. To remove the oscillation in the iteration process due to the nonsmoothness of the L 1/2 regularizer, we propose to smooth it in a neighborhood of the nonsmooth point to get a smoothing L 1/2 regularizer. By doing so, we expect to improve the efficiency of the L 1/2 regularizer so as to surpass the L 1 regularizer. Some of our recent works in this respect are summarized in this paper, including the works on BP feedforward neural networks, higher order neural networks, double parallel neural networks and Takagi-Sugeno fuzzy models.