A semi-supervised vector machine is a relatively new learning method using both labeled and unlabeled data in classifi- cation. Since the objective function of the model for an unstrained semi-supervised vector machin...A semi-supervised vector machine is a relatively new learning method using both labeled and unlabeled data in classifi- cation. Since the objective function of the model for an unstrained semi-supervised vector machine is not smooth, many fast opti- mization algorithms cannot be applied to solve the model. In order to overcome the difficulty of dealing with non-smooth objective functions, new methods that can solve the semi-supervised vector machine with desired classification accuracy are in great demand. A quintic spline function with three-times differentiability at the ori- gin is constructed by a general three-moment method, which can be used to approximate the symmetric hinge loss function. The approximate accuracy of the quintic spiine function is estimated. Moreover, a quintic spline smooth semi-support vector machine is obtained and the convergence accuracy of the smooth model to the non-smooth one is analyzed. Three experiments are performed to test the efficiency of the model. The experimental results show that the new model outperforms other smooth models, in terms of classification performance. Furthermore, the new model is not sensitive to the increasing number of the labeled samples, which means that the new model is more efficient.展开更多
Consider X and Y are two real or complex Banach spaces. We introduce and study a modified Gauss-type proximal point algorithm (in short modified G-PPA) for solving the generalized equations of the form 0 ∈h(...Consider X and Y are two real or complex Banach spaces. We introduce and study a modified Gauss-type proximal point algorithm (in short modified G-PPA) for solving the generalized equations of the form 0 ∈h(x) + H(x), where h : X → Y is a smooth function on Ω ⊆X and H : X ⇉2<sup>Y</sup> is a set valued mapping with closed graph. When H is metrically regular and under some sufficient conditions, we analyze both semi-local and local convergence of the modified G-PPA. Moreover, the convergence results of the modified G-PPA are justified by presenting a numerical example.展开更多
为了解决半监督支持向量分类优化模型中的非凸非光滑问题,基于分段逼近的思想提出了一个分段函数,并以此逼近非凸非光滑的目标函数。给出的分段函数可以根据不同的精度要求选择不同的逼近参数,同时构造出基于上述分段函数的光滑半监督...为了解决半监督支持向量分类优化模型中的非凸非光滑问题,基于分段逼近的思想提出了一个分段函数,并以此逼近非凸非光滑的目标函数。给出的分段函数可以根据不同的精度要求选择不同的逼近参数,同时构造出基于上述分段函数的光滑半监督支持向量机模型。采用了LDS(Low Density Separation)算法求解模型,分析了其对对称铰链损失函数的逼进精度。理论分析和数值实验结果都证明分段光滑的半监督支持向量机的分类性能和效率优于以往提出的光滑模型。展开更多
In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., th...In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.展开更多
We study the smoothing method for the solution of generalized semi-infinite optimization problems from （O. Stein, G. Still： Solving semi-infinite optimization problems with interior point techniques, SIAM J. Control...We study the smoothing method for the solution of generalized semi-infinite optimization problems from （O. Stein, G. Still： Solving semi-infinite optimization problems with interior point techniques, SIAM J. Control Optim., 42（2003）, pp. 769-788）. It is shown that Karush-Kuhn-Tucker points of the smoothed problems do not necessarily converge to a Karush-Kuhn-Tucker point of the original problem, as could be expected from results in （F. Facchinei, H. Jiang, L. Qi： A smoothing method for mathematical programs with equilibrium constraints, Math. Program., 85（1999）, pp. 107-134）. Instead, they might merely converge to a Fritz John point. We give, however, different additional assumptions which guarantee convergence to Karush-Kuhn：Tucker points.展开更多
基金supported by the Fundamental Research Funds for University of Science and Technology Beijing(FRF-BR-12-021)
文摘A semi-supervised vector machine is a relatively new learning method using both labeled and unlabeled data in classifi- cation. Since the objective function of the model for an unstrained semi-supervised vector machine is not smooth, many fast opti- mization algorithms cannot be applied to solve the model. In order to overcome the difficulty of dealing with non-smooth objective functions, new methods that can solve the semi-supervised vector machine with desired classification accuracy are in great demand. A quintic spline function with three-times differentiability at the ori- gin is constructed by a general three-moment method, which can be used to approximate the symmetric hinge loss function. The approximate accuracy of the quintic spiine function is estimated. Moreover, a quintic spline smooth semi-support vector machine is obtained and the convergence accuracy of the smooth model to the non-smooth one is analyzed. Three experiments are performed to test the efficiency of the model. The experimental results show that the new model outperforms other smooth models, in terms of classification performance. Furthermore, the new model is not sensitive to the increasing number of the labeled samples, which means that the new model is more efficient.
文摘Consider X and Y are two real or complex Banach spaces. We introduce and study a modified Gauss-type proximal point algorithm (in short modified G-PPA) for solving the generalized equations of the form 0 ∈h(x) + H(x), where h : X → Y is a smooth function on Ω ⊆X and H : X ⇉2<sup>Y</sup> is a set valued mapping with closed graph. When H is metrically regular and under some sufficient conditions, we analyze both semi-local and local convergence of the modified G-PPA. Moreover, the convergence results of the modified G-PPA are justified by presenting a numerical example.
文摘为了解决半监督支持向量分类优化模型中的非凸非光滑问题,基于分段逼近的思想提出了一个分段函数,并以此逼近非凸非光滑的目标函数。给出的分段函数可以根据不同的精度要求选择不同的逼近参数,同时构造出基于上述分段函数的光滑半监督支持向量机模型。采用了LDS(Low Density Separation)算法求解模型,分析了其对对称铰链损失函数的逼进精度。理论分析和数值实验结果都证明分段光滑的半监督支持向量机的分类性能和效率优于以往提出的光滑模型。
基金Supported by the National Natural Science Foundation of China(No.10671203,No.70621001) and the faculty research grant at MSU
文摘In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.
文摘We study the smoothing method for the solution of generalized semi-infinite optimization problems from （O. Stein, G. Still： Solving semi-infinite optimization problems with interior point techniques, SIAM J. Control Optim., 42（2003）, pp. 769-788）. It is shown that Karush-Kuhn-Tucker points of the smoothed problems do not necessarily converge to a Karush-Kuhn-Tucker point of the original problem, as could be expected from results in （F. Facchinei, H. Jiang, L. Qi： A smoothing method for mathematical programs with equilibrium constraints, Math. Program., 85（1999）, pp. 107-134）. Instead, they might merely converge to a Fritz John point. We give, however, different additional assumptions which guarantee convergence to Karush-Kuhn：Tucker points.
