It is a common practice to evaluate probability density function or matter spatial density function from statistical samples. Kernel density estimation is a frequently used method, but to select an optimal bandwidth o...It is a common practice to evaluate probability density function or matter spatial density function from statistical samples. Kernel density estimation is a frequently used method, but to select an optimal bandwidth of kernel estimation, which is completely based on data samples, is a long-term issue that has not been well settled so far. There exist analytic formulae of optimal kernel bandwidth, but they cannot be applied directly to data samples,since they depend on the unknown underlying density functions from which the samples are drawn. In this work, we devise an approach to pick out the totally data-based optimal bandwidth. First, we derive correction formulae for the analytic formulae of optimal bandwidth to compute the roughness of the sample's density function. Then substitute the correction formulae into the analytic formulae for optimal bandwidth, and through iteration we obtain the sample's optimal bandwidth. Compared with analytic formulae, our approach gives very good results, with relative differences from the analytic formulae being only 2%～3% for sample size larger than 10~4. This approach can also be generalized easily to cases of variable kernel estimations.展开更多
The state of health SoH of lithium ion batteries plays a predominant role in ensuring the safe and reliable operation of electric vehicles.In this,a novel SoH estimation approach using support vector regression with a...The state of health SoH of lithium ion batteries plays a predominant role in ensuring the safe and reliable operation of electric vehicles.In this,a novel SoH estimation approach using support vector regression with a Gaussian kernel optimized using the Bayesian optimization technique(BO-SVR with a Gaussian kernel)was proposed.Unlike,traditional approaches that use the internal resistance,and battery capacity as input parameters,this study utilized the equivalent discharging voltage difference interval and equivalent charging voltage difference interval,as they capture the dynamic voltage characteristics associated with the battery degradation.The model was simulated using MATLAB 2023a.The mean absolute error,R^(2),root mean squared error,and mean squared error were considered as performance indicators.The simulation results indicated that the proposed BO-SVR with a Gaussian kernel model had superior performance to other kernel SVR and Gaussian Process Regression models,with a reduced RMSE of 0.0082,thus demonstrating its potential to predict the SoH more accurately.展开更多
Silicone material extrusion(MEX)is widely used for processing liquids and pastes.Owing to the uneven linewidth and elastic extrusion deformation caused by material accumulation,products may exhibit geometric errors an...Silicone material extrusion(MEX)is widely used for processing liquids and pastes.Owing to the uneven linewidth and elastic extrusion deformation caused by material accumulation,products may exhibit geometric errors and performance defects,leading to a decline in product quality and affecting its service life.This study proposes a process parameter optimization method that considers the mechanical properties of printed specimens and production costs.To improve the quality of silicone printing samples and reduce production costs,three machine learning models,kernel extreme learning machine(KELM),support vector regression(SVR),and random forest(RF),were developed to predict these three factors.Training data were obtained through a complete factorial experiment.A new dataset is obtained using the Euclidean distance method,which assigns the elimination factor.It is trained with Bayesian optimization algorithms for parameter optimization,the new dataset is input into the improved double Gaussian extreme learning machine,and finally obtains the improved KELM model.The results showed improved prediction accuracy over SVR and RF.Furthermore,a multi-objective optimization framework was proposed by combining genetic algorithm technology with the improved KELM model.The effectiveness and reasonableness of the model algorithm were verified by comparing the optimized results with the experimental results.展开更多
By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent ...By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent conditions of the optimal matching parameter are established,and the expression of the optimal constant factor is obtained.Finally,their applications in operator theory are considered.展开更多
In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barr...In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barrier term. Iteration bounds both for large-and small-update methods are derived, namely, O（nlog（n/c）） and O（√nlog（n/ε））. This new kernel function has simple algebraic expression and the proximity function has not been used before. Analogous to the classical logarithmic kernel function, our complexity analysis is easier than the other pri- mal-dual interior-point methods based on logarithmic barrier functions and recent kernel functions.展开更多
A robust low-carbon economic optimal scheduling method that considers source-load uncertainty and hydrogen energy utilization is developed.The proposed method overcomes the challenge of source-load random fluctuations...A robust low-carbon economic optimal scheduling method that considers source-load uncertainty and hydrogen energy utilization is developed.The proposed method overcomes the challenge of source-load random fluctuations in integrated energy systems(IESs)in the operation scheduling problem of integrated energy production units(IEPUs).First,to solve the problem of inaccurate prediction of renewable energy output,an improved robust kernel density estimation method is proposed to construct a data-driven uncertainty output set of renewable energy sources statistically and build a typical scenario of load uncertainty using stochastic scenario reduction.Subsequently,to resolve the problem of insufficient utilization of hydrogen energy in existing IEPUs,a robust low-carbon economic optimal scheduling model of the source-load interaction of an IES with a hydrogen energy system is established.The system considers the further utilization of energy using hydrogen energy coupling equipment(such as hydrogen storage devices and fuel cells)and the comprehensive demand response of load-side schedulable resources.The simulation results show that the proposed robust stochastic optimization model driven by data can effectively reduce carbon dioxide emissions,improve the source-load interaction of the IES,realize the efficient use of hydrogen energy,and improve system robustness.展开更多
In this paper,we propose and analyze a full-Newton step feasible interior-point algorithm for semidefinite optimization based on a kernel function with linear growth term.The kernel function is used both for determini...In this paper,we propose and analyze a full-Newton step feasible interior-point algorithm for semidefinite optimization based on a kernel function with linear growth term.The kernel function is used both for determining the search directions and for measuring the distance between the given iterate and theμ-center for the algorithm.By developing a new norm-based proximity measure and some technical results,we derive the iteration bound that coincides with the currently best known iteration bound for the algorithm with small-update method.In our knowledge,this result is the first instance of full-Newton step feasible interior-point method for SDO which involving the kernel function.展开更多
This paper addresses the high dimension sample problem in discriminate analysis under nonparametric and supervised assumptions. Since there is a kind of equivalence between the probabilistic dependence measure and the...This paper addresses the high dimension sample problem in discriminate analysis under nonparametric and supervised assumptions. Since there is a kind of equivalence between the probabilistic dependence measure and the Bayes classification error probability, we propose to use an iterative algorithm to optimize the dimension reduction for classification with a probabilistic approach to achieve the Bayes classifier. The estimated probabilities of different errors encountered along the different phases of the system are realized by the Kernel estimate which is adjusted in a means of the smoothing parameter. Experiment results suggest that the proposed approach performs well.展开更多
Reducing pollutant emissions from electricity production in the power system positively impacts the control of greenhouse gas emissions.Boosting kernel search optimizer(BKSO)is introduced in this research to solve the...Reducing pollutant emissions from electricity production in the power system positively impacts the control of greenhouse gas emissions.Boosting kernel search optimizer(BKSO)is introduced in this research to solve the combined economic emission dispatch(CEED)problem.Inspired by the foraging behavior in the slime mould algorithm(SMA),the kernel matrix of the kernel search optimizer(KSO)is intensified.The proposed BKSO is superior to the standard KSO in terms of exploitation ability,robustness,and convergence rate.The CEC2013 test function is used to assess the improved KSO's performance and compared to 11 well-known optimization algorithms.BKSO performs better in statistical results and convergence curves.At the same time,BKSO achieves better fuel costs and fewer pollution emissions by testing with four real CEED cases,and the Pareto solution obtained is also better than other MAs.Based on the experimental results,BKSO has better performance than other comparable MAs and can provide more economical,robust,and cleaner solutions to CEED problems.展开更多
Power transformer is one of the most crucial devices in power grid.It is significant to determine incipient faults of power transformers fast and accurately.Input features play critical roles in fault diagnosis accura...Power transformer is one of the most crucial devices in power grid.It is significant to determine incipient faults of power transformers fast and accurately.Input features play critical roles in fault diagnosis accuracy.In order to further improve the fault diagnosis performance of power trans-formers,a random forest feature selection method coupled with optimized kernel extreme learning machine is presented in this study.Firstly,the random forest feature selection approach is adopted to rank 42 related input features derived from gas concentration,gas ratio and energy-weighted dissolved gas analysis.Afterwards,a kernel extreme learning machine tuned by the Aquila optimization algorithm is implemented to adjust crucial parameters and select the optimal feature subsets.The diagnosis accuracy is used to assess the fault diagnosis capability of concerned feature subsets.Finally,the optimal feature subsets are applied to establish fault diagnosis model.According to the experimental results based on two public datasets and comparison with 5 conventional approaches,it can be seen that the average accuracy of the pro-posed method is up to 94.5%,which is superior to that of other conventional approaches.Fault diagnosis performances verify that the optimum feature subset obtained by the presented method can dramatically improve power transformers fault diagnosis accuracy.展开更多
Panicle swarm optimization （PSO） is an optimization algorithm based on the swarm intelligent principle. In this paper the modified PSO is applied to a kernel principal component analysis （ KPCA ） for an optimal ke...Panicle swarm optimization （PSO） is an optimization algorithm based on the swarm intelligent principle. In this paper the modified PSO is applied to a kernel principal component analysis （ KPCA ） for an optimal kernel function parameter. We first comprehensively considered within-class scatter and between-class scatter of the sample features. Then, the fitness function of an optimized kernel function parameter is constructed, and the particle swarm optimization algorithm with adaptive acceleration （CPSO） is applied to optimizing it. It is used for gearbox condi- tion recognition, and the result is compared with the recognized results based on principal component analysis （PCA）. The results show that KPCA optimized by CPSO can effectively recognize fault conditions of the gearbox by reducing bind set-up of the kernel function parameter, and its results of fault recognition outperform those of PCA. We draw the conclusion that KPCA based on CPSO has an advantage in nonlinear feature extraction of mechanical failure, and is helpful for fault condition recognition of complicated machines.展开更多
The precision of the kernel independent component analysis( KICA) algorithm depends on the type and parameter values of kernel function. Therefore,it's of great significance to study the choice method of KICA'...The precision of the kernel independent component analysis( KICA) algorithm depends on the type and parameter values of kernel function. Therefore,it's of great significance to study the choice method of KICA's kernel parameters for improving its feature dimension reduction result. In this paper, a fitness function was established by use of the ideal of Fisher discrimination function firstly. Then the global optimal solution of fitness function was searched by particle swarm optimization( PSO) algorithm and a multi-state information dimension reduction algorithm based on PSO-KICA was established. Finally,the validity of this algorithm to enhance the precision of feature dimension reduction has been proven.展开更多
In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure betwe...In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity O（ √r（1/2）（log r）^2 log（r/ ε））, which is as good as the convex quadratic semi-definite optimization analogue.展开更多
滚动轴承是机械设备中的常见关键部件,准确预测其剩余使用寿命对机械设备的安全稳定运行至关重要。针对目前轴承寿命预测存在的轴承退化特征不明显、模型泛化能力差以及数据长期依赖关系难以捕捉的问题,提出基于时频域信号优化器(Time-F...滚动轴承是机械设备中的常见关键部件,准确预测其剩余使用寿命对机械设备的安全稳定运行至关重要。针对目前轴承寿命预测存在的轴承退化特征不明显、模型泛化能力差以及数据长期依赖关系难以捕捉的问题,提出基于时频域信号优化器(Time-Frequency domain signal Ratio Optimizer,TFRO)的多重膨胀多核时间卷积网络(Multi inflated Multi kernel Time Convolutional Network,Mi-MkTCN)模型。TFRO优化器为了精准记忆重要信息,在每一个时间节点上,将过去信息和当前信息重组,其中过去信息中的重要的时频域特征经过了有比例的分配。Mi-MkTCN利用多重膨胀确保重要特征不丢失,再利用多核时间卷积网络实现对不同尺度特征的提取。最终的消融对比实验验证了改进方法的有效性,模型的平均绝对误差、均方误差及均方根误差指标分别为0.00145、0.05069和0.12045。实验结果表明,所提方法显著提升了轴承剩余使用寿命的预测精度,为轴承剩余使用寿命预测提供了高精度、高鲁棒性的解决方案。展开更多
基金Supported by the National Science Foundation of China under Grant No.11273013by the Natural Science Foundation of Jilin Province under Grant No.20180101228JC
文摘It is a common practice to evaluate probability density function or matter spatial density function from statistical samples. Kernel density estimation is a frequently used method, but to select an optimal bandwidth of kernel estimation, which is completely based on data samples, is a long-term issue that has not been well settled so far. There exist analytic formulae of optimal kernel bandwidth, but they cannot be applied directly to data samples,since they depend on the unknown underlying density functions from which the samples are drawn. In this work, we devise an approach to pick out the totally data-based optimal bandwidth. First, we derive correction formulae for the analytic formulae of optimal bandwidth to compute the roughness of the sample's density function. Then substitute the correction formulae into the analytic formulae for optimal bandwidth, and through iteration we obtain the sample's optimal bandwidth. Compared with analytic formulae, our approach gives very good results, with relative differences from the analytic formulae being only 2%～3% for sample size larger than 10~4. This approach can also be generalized easily to cases of variable kernel estimations.
基金supported by the Royal Academy of Engineering,UK,under the scheme of Distinguished International Associates(DIA-2424-5-134).
文摘The state of health SoH of lithium ion batteries plays a predominant role in ensuring the safe and reliable operation of electric vehicles.In this,a novel SoH estimation approach using support vector regression with a Gaussian kernel optimized using the Bayesian optimization technique(BO-SVR with a Gaussian kernel)was proposed.Unlike,traditional approaches that use the internal resistance,and battery capacity as input parameters,this study utilized the equivalent discharging voltage difference interval and equivalent charging voltage difference interval,as they capture the dynamic voltage characteristics associated with the battery degradation.The model was simulated using MATLAB 2023a.The mean absolute error,R^(2),root mean squared error,and mean squared error were considered as performance indicators.The simulation results indicated that the proposed BO-SVR with a Gaussian kernel model had superior performance to other kernel SVR and Gaussian Process Regression models,with a reduced RMSE of 0.0082,thus demonstrating its potential to predict the SoH more accurately.
基金supported by the National Key R&D Program of China(No.2022YFA1005204l)。
文摘Silicone material extrusion(MEX)is widely used for processing liquids and pastes.Owing to the uneven linewidth and elastic extrusion deformation caused by material accumulation,products may exhibit geometric errors and performance defects,leading to a decline in product quality and affecting its service life.This study proposes a process parameter optimization method that considers the mechanical properties of printed specimens and production costs.To improve the quality of silicone printing samples and reduce production costs,three machine learning models,kernel extreme learning machine(KELM),support vector regression(SVR),and random forest(RF),were developed to predict these three factors.Training data were obtained through a complete factorial experiment.A new dataset is obtained using the Euclidean distance method,which assigns the elimination factor.It is trained with Bayesian optimization algorithms for parameter optimization,the new dataset is input into the improved double Gaussian extreme learning machine,and finally obtains the improved KELM model.The results showed improved prediction accuracy over SVR and RF.Furthermore,a multi-objective optimization framework was proposed by combining genetic algorithm technology with the improved KELM model.The effectiveness and reasonableness of the model algorithm were verified by comparing the optimized results with the experimental results.
基金Supported by National Natural Science Foundation of China(Grant No.12071491)Guangzhou Science and Technology Plan Project(Grant No.202102080177).
文摘By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent conditions of the optimal matching parameter are established,and the expression of the optimal constant factor is obtained.Finally,their applications in operator theory are considered.
基金Supported by the Natural Science Foundation of Hubei Province (2008CDZD47)
文摘In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barrier term. Iteration bounds both for large-and small-update methods are derived, namely, O（nlog（n/c）） and O（√nlog（n/ε））. This new kernel function has simple algebraic expression and the proximity function has not been used before. Analogous to the classical logarithmic kernel function, our complexity analysis is easier than the other pri- mal-dual interior-point methods based on logarithmic barrier functions and recent kernel functions.
基金supported by the National Key Research and Development Project of China(2018YFE0122200).
文摘A robust low-carbon economic optimal scheduling method that considers source-load uncertainty and hydrogen energy utilization is developed.The proposed method overcomes the challenge of source-load random fluctuations in integrated energy systems(IESs)in the operation scheduling problem of integrated energy production units(IEPUs).First,to solve the problem of inaccurate prediction of renewable energy output,an improved robust kernel density estimation method is proposed to construct a data-driven uncertainty output set of renewable energy sources statistically and build a typical scenario of load uncertainty using stochastic scenario reduction.Subsequently,to resolve the problem of insufficient utilization of hydrogen energy in existing IEPUs,a robust low-carbon economic optimal scheduling model of the source-load interaction of an IES with a hydrogen energy system is established.The system considers the further utilization of energy using hydrogen energy coupling equipment(such as hydrogen storage devices and fuel cells)and the comprehensive demand response of load-side schedulable resources.The simulation results show that the proposed robust stochastic optimization model driven by data can effectively reduce carbon dioxide emissions,improve the source-load interaction of the IES,realize the efficient use of hydrogen energy,and improve system robustness.
基金Supported by University Science Research Project of Anhui Province(KJ2019A1297)University Teaching Research Project of Anhui Province(2019jxtd144)。
文摘In this paper,we propose and analyze a full-Newton step feasible interior-point algorithm for semidefinite optimization based on a kernel function with linear growth term.The kernel function is used both for determining the search directions and for measuring the distance between the given iterate and theμ-center for the algorithm.By developing a new norm-based proximity measure and some technical results,we derive the iteration bound that coincides with the currently best known iteration bound for the algorithm with small-update method.In our knowledge,this result is the first instance of full-Newton step feasible interior-point method for SDO which involving the kernel function.
文摘This paper addresses the high dimension sample problem in discriminate analysis under nonparametric and supervised assumptions. Since there is a kind of equivalence between the probabilistic dependence measure and the Bayes classification error probability, we propose to use an iterative algorithm to optimize the dimension reduction for classification with a probabilistic approach to achieve the Bayes classifier. The estimated probabilities of different errors encountered along the different phases of the system are realized by the Kernel estimate which is adjusted in a means of the smoothing parameter. Experiment results suggest that the proposed approach performs well.
基金This research was supported by the Science&Technology Development Project of Jilin Province,China(YDZJ202201ZYTS555)the Science&Technology Research Project of the Education Department of Jilin Province,China(JJKH20220244KJ)。
文摘Reducing pollutant emissions from electricity production in the power system positively impacts the control of greenhouse gas emissions.Boosting kernel search optimizer(BKSO)is introduced in this research to solve the combined economic emission dispatch(CEED)problem.Inspired by the foraging behavior in the slime mould algorithm(SMA),the kernel matrix of the kernel search optimizer(KSO)is intensified.The proposed BKSO is superior to the standard KSO in terms of exploitation ability,robustness,and convergence rate.The CEC2013 test function is used to assess the improved KSO's performance and compared to 11 well-known optimization algorithms.BKSO performs better in statistical results and convergence curves.At the same time,BKSO achieves better fuel costs and fewer pollution emissions by testing with four real CEED cases,and the Pareto solution obtained is also better than other MAs.Based on the experimental results,BKSO has better performance than other comparable MAs and can provide more economical,robust,and cleaner solutions to CEED problems.
基金support of national natural science foundation of China(No.52067021)natural science foundation of Xinjiang(2022D01C35)+1 种基金excellent youth scientific and technological talents plan of Xinjiang(No.2019Q012)major science and technology special project of Xinjiang Uygur Autonomous Region(2022A01002-2).
文摘Power transformer is one of the most crucial devices in power grid.It is significant to determine incipient faults of power transformers fast and accurately.Input features play critical roles in fault diagnosis accuracy.In order to further improve the fault diagnosis performance of power trans-formers,a random forest feature selection method coupled with optimized kernel extreme learning machine is presented in this study.Firstly,the random forest feature selection approach is adopted to rank 42 related input features derived from gas concentration,gas ratio and energy-weighted dissolved gas analysis.Afterwards,a kernel extreme learning machine tuned by the Aquila optimization algorithm is implemented to adjust crucial parameters and select the optimal feature subsets.The diagnosis accuracy is used to assess the fault diagnosis capability of concerned feature subsets.Finally,the optimal feature subsets are applied to establish fault diagnosis model.According to the experimental results based on two public datasets and comparison with 5 conventional approaches,it can be seen that the average accuracy of the pro-posed method is up to 94.5%,which is superior to that of other conventional approaches.Fault diagnosis performances verify that the optimum feature subset obtained by the presented method can dramatically improve power transformers fault diagnosis accuracy.
基金supported by National Natural Science Foundation under Grant No.50875247Shanxi Province Natural Science Foundation under Grant No.2009011026-1
文摘Panicle swarm optimization （PSO） is an optimization algorithm based on the swarm intelligent principle. In this paper the modified PSO is applied to a kernel principal component analysis （ KPCA ） for an optimal kernel function parameter. We first comprehensively considered within-class scatter and between-class scatter of the sample features. Then, the fitness function of an optimized kernel function parameter is constructed, and the particle swarm optimization algorithm with adaptive acceleration （CPSO） is applied to optimizing it. It is used for gearbox condi- tion recognition, and the result is compared with the recognized results based on principal component analysis （PCA）. The results show that KPCA optimized by CPSO can effectively recognize fault conditions of the gearbox by reducing bind set-up of the kernel function parameter, and its results of fault recognition outperform those of PCA. We draw the conclusion that KPCA based on CPSO has an advantage in nonlinear feature extraction of mechanical failure, and is helpful for fault condition recognition of complicated machines.
文摘The precision of the kernel independent component analysis( KICA) algorithm depends on the type and parameter values of kernel function. Therefore,it's of great significance to study the choice method of KICA's kernel parameters for improving its feature dimension reduction result. In this paper, a fitness function was established by use of the ideal of Fisher discrimination function firstly. Then the global optimal solution of fitness function was searched by particle swarm optimization( PSO) algorithm and a multi-state information dimension reduction algorithm based on PSO-KICA was established. Finally,the validity of this algorithm to enhance the precision of feature dimension reduction has been proven.
基金Supported by the Natural Science Foundation of Hubei Province(2008CDZD47)
文摘In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity O（ √r（1/2）（log r）^2 log（r/ ε））, which is as good as the convex quadratic semi-definite optimization analogue.
文摘滚动轴承是机械设备中的常见关键部件,准确预测其剩余使用寿命对机械设备的安全稳定运行至关重要。针对目前轴承寿命预测存在的轴承退化特征不明显、模型泛化能力差以及数据长期依赖关系难以捕捉的问题,提出基于时频域信号优化器(Time-Frequency domain signal Ratio Optimizer,TFRO)的多重膨胀多核时间卷积网络(Multi inflated Multi kernel Time Convolutional Network,Mi-MkTCN)模型。TFRO优化器为了精准记忆重要信息,在每一个时间节点上,将过去信息和当前信息重组,其中过去信息中的重要的时频域特征经过了有比例的分配。Mi-MkTCN利用多重膨胀确保重要特征不丢失,再利用多核时间卷积网络实现对不同尺度特征的提取。最终的消融对比实验验证了改进方法的有效性,模型的平均绝对误差、均方误差及均方根误差指标分别为0.00145、0.05069和0.12045。实验结果表明,所提方法显著提升了轴承剩余使用寿命的预测精度,为轴承剩余使用寿命预测提供了高精度、高鲁棒性的解决方案。
