非线性支持向量分类机解的二阶充分条件

优化问题解的二阶充分条件是研究其灵敏度分析的基础,支持向量分类机是新的数据挖掘优化问题.给出了支持向量分类机的解满足二阶充分条件成立定理;定理的假设条件是很弱的,用支持向量分类机求解实际问题,通常总假定这一条件成立;特别地,对线性可分支持向量机问题,其解满足二阶充分条件成为当然成立的事实.     

线性ε-支持向量回归机的数据扰动分析

支持向量回归机是解决回归问题的一个重要方法.在实际问题中由于测量及计算误差的存在,我们得到的数据往往只是真值的某种近似,带有一定的舍入误差,因此有必要研究支持向量回归机的数据扰动问题.考虑到线性回归问题在实际生活中有广泛的应用价值,把线性ε-支持向量回归机作为研究对象.由于最终关心的是它的原始问题的解,所以我们研究给定的训练集中输入数据发生微小地扰动后,原始问题的解的变化情况.在一定的条件下给出了解对扰动数据偏导数的表达式,建立了线性ε-支持向量回归机的原始问题的灵敏度分析定理.文中还进一步分析了建立该灵敏度分析定理所需要的条件,给出了条件减弱后的结果.文章最后还通过一些简单的数值试验验证了定理的准确性.     

矩阵加权QR分解的严格扰动界

矩阵的加权QR分解为矩阵的QR分解的推广,可以用来求解加权线性最小二乘问题.本文利用矩阵方程方法与修正的矩阵方程方法相结合的方法及修正的矩阵-向量方程方法与Lyapunov控制函数和Banach不动点定理相结合的方法获得加权QR分解在范数型扰动下的范数型的严格扰动界.     

可靠性分析中的最小二乘支持向量机分类方法

为了提高支持向量分类机在处理大样本可靠性问题时的计算效率,将最小二乘支持向量分类机引入到可靠性分析中,使得支持向量机中的二次规划问题转化为求解线性方程组问题,减少了计算量.数值算例表明:基于最小二乘支持向量分类机的可靠性方法与基于支持向量分类机的可靠性方法具有一样的计算精度,而且前者的计算效率明显优于后者.     

基于Fuzzy理论的数据挖掘算法研究(Ⅰ)

“数据挖掘”是数据处理的一个新领域.支持向量机是数据挖掘的一种新方法,该技术在很多领域得到了成功的应用.但是,支持向量机目前还存在许多局限,当支持向量机的训练集中含有模糊信息时,支持向量机将无能为力.为解决一般情况下支持向量机中含有模糊信息(模糊参数)问题,研究了模糊机会约束规划、模糊分类中的模糊特征及其表示方法,建立了模糊支持向量分类机理论,给出了模糊线性可分的模糊支持向量分类机算法.     

基于Fuzzy理论的数据挖掘算法研究Ⅱ

"数据挖掘"是数据处理的一个新领域.支持向量机是数据挖掘的一种新方法,该技术在很多领域得到了成功的应用.但是,支持向量机目前还存在许多局限,当支持向量机的训练集中含有模糊信息时,支持向量机将无能为力.为解决一般情况下支持向量机中含有模糊信息(模糊参数)问题,研究了模糊机会约束规划、模糊分类中的模糊特征及其表示方法,建立了模糊支持向量分类机理论,给出了模糊线性可分的模糊支持向量分类机算法.     

支持向量分类方法理论基础的改进

支持向量机是通过求解对偶问题来解决原始问题的.针对线性决策函数f(x)=(w·x)+b,我们指出了其原有的逻辑系统中的错误,并通过严格的证明,对其理论基础作了改进.而且,对于阈值b,我们给出了一个新的简洁计算公式.     

基于KMOD核函数的SVM方法在信用评分中的应用

本文介绍了支持向量分类机,并引入具有更好识别能力的KMOD核函数建立了SVM信用卡分类模型.利用澳大利亚和德国的信用卡数据进行了数值实验,结果表明该模型在分类准确率、支持向量方面优于基于RBF的SVM模型.     

基于支持向量机的神经元形态分类

针对神经元的空间几何形态特征分类问题以及神经元的生长预测问题进行了探讨.结合神经元的形态数据,分别建立了基于支持向量机的神经元形态分类模型、基于主成分分析和支持向量机的神经元分类模型以及基于遗传算法和RBF网络的神经元生长预测模型,在较合理的假设下,对各个模型进行求解,得到了较理想的结果.     

新的非线性分离定理及其在向量优化中的应用

本文利用Minkowski型非线性标量化泛函分别建立了一般实线性空间中基于相对代数内部与向量闭包,实拓扑线性空间中基于相对拓扑内部与拓扑闭包,以及实分离局部凸拓扑线性空间中基于拟相对内部与拓扑闭包的非线性分离定理.这些新的分离定理能够用于研究序锥的拓扑内部甚至是相对拓扑内部或相对代数内部可能为空的向量优化问题.作为其应用,本文给出了向量优化问题相应弱有效解的一些非线性标量化性质;此外,也提出了无限维空间中的一些具体例子来对主要结果进行了解释.     

Robust conjugate duality for convex optimization under uncertainty with application to data classification

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.     

Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems

The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Fréchet differentiability of all operators involved, and the existence of the topological complement of the null space of the Fréchet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.The authors are indebted to the Natural Sciences and Engineering Research Council of Canada for financial support through Grants A4109 and A7329, respectively. They would also like to thank the referee for his comments.     

Perturbation Approach to Sensitivity Analysis in Mathematical Programming

This paper presents a perturbation approach for performing sensitivity analysis of mathematical programming problems. Contrary to standard methods, the active constraints are not assumed to remain active if the problem data are perturbed, nor the partial derivatives are assumed to exist. In other words, all the elements, variables, parameters, Karush–Kuhn–Tucker multipliers, and objective function values may vary provided that optimality is maintained and the general structure of a feasible perturbation (which is a polyhedral cone) is obtained. This allows determining: (a) the local sensitivities, (b) whether or not partial derivatives exist, and (c) if the directional derivative for a given direction exists. A method for the simultaneous obtention of the sensitivities of the objective function optimal value and the primal and dual variable values with respect to data is given. Three examples illustrate the concepts presented and the proposed methodology. Finally, some relevant conclusions are drawn. The authors are indebted to the Ministry of Science and Education of Spain, Projects CICYT DPI2002-04172-C04-02 and CICYT DPI2003-01362, and to the Fulbright Commission for partial support. The authors are grateful to the referees for comments improving the quality of the paper.     

On the optimal value function for certain linear programs with unbounded optimal solution sets

Often, the coefficients of a linear programming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.The author would like to thank two anonymous reviewers for their most useful comments on earlier versions of this paper.     

Sensitivity analysis of maximally monotone inclusions via the proto-differentiability of the resolvent operator

This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account. Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative of the resolvent operator associated to the maximally monotone parameterized variational inclusion. This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being itself a solution of a variational inclusion involving the semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual composite monotone inclusion is given. Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their derivatives belong to the derivative of the associated Kuhn–Tucker set.

     

基于支持向量机混合采样的不平衡数据分类方法

利用传统支持向量机(SVM)对不平衡数据进行分类时,由于真实的少数类支持向量样本过少且难以被识别,造成了分类时效果不是很理想.针对这一问题,提出了一种基于支持向量机混合采样的不平衡数据分类方法(BSMS).该方法首先对经过支持向量机分类的原始不平衡数据按照所处位置的不同划分为支持向量区(SV),多数类非支持向量区(MNSV)以及少数类非支持向量区(FNSV)三个区域,并对MNSV区和FNSV区的样本做去噪处理;然后对SV区分类错误和部分分类正确且靠近决策边界的少数类样本重复进行过采样处理,直到找到测试结果最优的训练数据集;最后有选择的随机删除MNSV区的部分样本.实验结果表明:方法优于其他采样方法.     

线性支持向量顺序回归机的原始问题的解集分析

本文主要对线性支持向量顺序回归机进行理论研究.对其相应原始问题解的存在性唯一性问题进行细致的分析,指明其解集的确切结构,并给出由对偶问题的解求出原始问题的解集的具体步骤.从而为建立理论上完备的线性支持向量顺序回归机提供了依据.     

Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation

This paper deals with the numerical analysis of time dependent parabolic partial differential equation. The equation has bistable nonlinearity and models electrical activity in a neuron. A qualitative analysis of the model is performed by means of a singular perturbation theory. A small parameter is introduced in the highest order derivative term. This small parameter is known as singular perturbation parameter. Boundary layers occur in the solution of singularly perturbed problems when the singular perturbation parameter tend to zero. These boundary layers are located in neighbourhoods of the boundary of the domain, where the solution has a very steep gradient. Most of the conventional methods fails to capture this effect. A numerical scheme is constructed to overcome this discrepancy in literature. A rigorous analysis is carried out to obtain a-priori estimates on the solution of the problem and its derivatives. It is then proven that the numerical method is unconditionally stable. Convergence and stability analysis is carried out. A set of numerical experiment is carried out and it is observed that the scheme faithfully mimics the dynamics of the model.     

A memetic approach to construct transductive discrete support vector machines

Transductive learning involves the construction and application of prediction models to classify a fixed set of decision objects into discrete groups. It is a special case of classification analysis with important applications in web-mining, corporate planning and other areas. This paper proposes a novel transductive classifier that is based on the philosophy of discrete support vector machines. We formalize the task to estimate the class labels of decision objects as a mixed integer program. A memetic algorithm is developed to solve the mathematical program and to construct a transductive support vector machine classifier, respectively. Empirical experiments on synthetic and real-world data evidence the effectiveness of the new approach and demonstrate that it identifies high quality solutions in short time. Furthermore, the results suggest that the class predictions following from the memetic algorithm are significantly more accurate than the predictions of a CPLEX-based reference classifier. Comparisons to other transductive and inductive classifiers provide further support for our approach and suggest that it performs competitive with respect to several benchmarks.