关于一类变分不等式的LQP算法

本文研究一类ξ-单调的变分不等式问题.利用KKT条件将原问题转换为非线性互补问题(nonlinear complementarity problem,NCP)的方法,获得了基于logarithmic-quadratic proximal(LQP)的算法及其改进形式,推广了LQP算法的适用范围.     

论求解单调变分不等式的一些投影收缩算法

论求解单调变分不等式的一些投影收缩算法何炳生（南京大学）ＯＮＳＯＭＥＰＲＯＪＥＣＴＩＯＮＡＮＤＣＯＮＴＲＡＣＴＩＯＮＭＥＴＨＯＤＳＦＯＲＳＯＬＶＩＮＧＭＯＮＯＴＯＮＥＶＡＲＩＡＴＩＯＮＡＬＩＮＥＱＵＡＬＩＴＩＥＳ￥ＨｅＢｉｎｇ－ｓｈｅｎｇ（Ｎａｍｉｎ...     

具不等式约束变分不等式的信赖域算法

1　引　　言令Ｘ是Ｒｎ 中的非空闭凸集 ,Ｆ :Ｘ→Ｒｎ 是连续映射 ,〈· ,·〉表示Ｒｎ 中的内积 有限维变分不等式问题 (以下简称变分不等式问题 ,记为ＶＩＰ或ＶＩ(Ｘ ,Ｆ) ) :就是求ｘ ∈Ｒｎ,使ｘ ∈Ｘ且 ｘ ∈Ｘ ,〈Ｆ(ｘ ) ,ｘ -ｘ 〉≥ 0 . ( 1 )在Ｘ =Ｒｎ+ 的特殊情形下 ,( 1 )变为非线性互补问题 (记为ＮＣＰ或ＮＣＰ(Ｆ) ) :就是求ｘ ∈Ｒｎ,使ｘ ≥ 0 ,Ｆ(ｘ ) ≥ 0 ,且〈ｘ ,Ｆ(ｘ )〉 =0 . ( 2 )　　变分不等式长期以来一直用于阐述和研究经济学、控制论、交通运输等领域中出现的各种平衡模型 近二十年来 ,变分不等式及其…     

求解单调变分不等式的两类迭代算法

给出了求解单调变分不等式的两类迭代算法．通过解强单调变分不等式子问题,产生两个迭代点列,都弱收敛到变分不等式的解．最后,给出了这两类新算法的收敛性分析．     

求解伪单调变分不等式的两种投影算法

本文提出了两种求解伪单调变分不等式的定步长的投影算法．这与Solodov ＆ Tseng(1996)和He(1997)的变步长策略不同．我们证明了算法的全局收敛性，并且还在一定条件下证明了算法的Q-线性收敛性．     

变分不等式问题的全局收敛性算法

本文给出了带一般凸约束的变分不等式问题的算法,并在多种线性搜索下证明了算法的全局收敛性。     

关于广义变分不等式与广义拟变分不等式

本文在非常一般的框架下,建立了极大极小不等式,广义变分不等式和广义拟变分不等式,证明了解的存在定理,且它们是在非紧集上得到的,从而推广和改进了[3～13]中的相应结果.     

Gwinner变分不等式和隐变分不等式

本文在研究Gwinner变分不等式的基础上，利用新的集值映象不动点定理，探讨一类具广泛意义的隐变分不等式问题，改进了迄今相关结果中对紧性条件的要求，在非紧设置下获得了解的存在性定理．     

一类线性约束变分不等式问题的幂罚函数法

本文研究了一类线性约束变分不等式(Ⅵ)的幂罚函数法求解问题.利用Ⅵ的KKT条件,将Ⅵ转化为等价的混合互补问题和一个新的Ⅵ问题,并在一定条件下分析了解的存在性和唯一性.利用度理论证明了幂罚方程组解的存在性与唯一性.由以上结果最终证明了幂罚函数法的收敛性,即幂罚方程组的解收敛于Ⅵ问题的解.     

关于一类随机变分不等式和随机拟变分不等式问题

本文对单值和多值情形的随机变分不等式和随机拟变分不等式得出可测解的存在性条件。另外还利用KKM-技巧及著名的Ky Fan定理对一类确定型的广义拟变分不等式讨论了解的存在性问题。本文的结果改进和发展了[10,11,12]中的重要结果。     

抛物型变分不等式解的支集的瞬间收缩性

研究了下面的抛物型变分不等式v≥0,(ut-Δu+b(x,t)up)(v-u)≥f(v-u)a.e.,(x,t)∈RN×(0,T],u≥0,(x,t)∈RN×(0,T],u(x,0)=u0(x),x∈RN的解的存在惟一性,以及解的支集的瞬间收缩性.     

Improved twin support vector machine

We improve the twin support vector machine(TWSVM)to be a novel nonparallel hyperplanes classifier,termed as ITSVM(improved twin support vector machine),for binary classification.By introducing the diferent Lagrangian functions for the primal problems in the TWSVM,we get an improved dual formulation of TWSVM,then the resulted ITSVM algorithm overcomes the common drawbacks in the TWSVMs and inherits the essence of the standard SVMs.Firstly,ITSVM does not need to compute the large inverse matrices before training which is inevitable for the TWSVMs.Secondly,diferent from the TWSVMs,kernel trick can be applied directly to ITSVM for the nonlinear case,therefore nonlinear ITSVM is superior to nonlinear TWSVM theoretically.Thirdly,ITSVM can be solved efciently by the successive overrelaxation(SOR)technique or sequential minimization optimization(SMO)method,which makes it more suitable for large scale problems.We also prove that the standard SVM is the special case of ITSVM.Experimental results show the efciency of our method in both computation time and classification accuracy.     

Scalarization and decomposition of vector variational inequalities governed by bifunctions

In this article we study the structure of solution sets within a special class of generalized Stampacchia-type vector variational inequalities, defined by means of a bifunction which takes values in a partially ordered Euclidean space. It is shown that, similar to multicriteria optimization problems, under appropriate convexity assumptions, the (weak) solutions of these vector variational inequalities can be recovered by solving a family of weighted scalar variational inequalities. Consequently, it is deduced that the set of weak solutions can be decomposed into the union of the sets of strong solutions of all variational inequalities obtained from the original one by selecting certain components of the bifunction which governs it.     

Equilibrium problems and variational inequalities in topological vector spaces

An existence result for the equilibrium problem is proved in a general topological vector space. As applications, existence results are derived for variational inequality problems, vector equilibrium problems and vector variational inequality problems. Our results extend and unify a number of existence theorems in non-compact cases     

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.     

On the nonemptiness and compactness of the solution sets for vector variational inequalities

In this article, we investigate conditions for nonemptiness and compactness of the sets of solutions of pseudomonotone vector variational inequalities by using the concept of asymptotical cones. We show that a pseudomonotone vector variational inequality has a nonempty and compact solution set provided that it is strictly feasible. We also obtain some necessary conditions for the set of solutions of a pseudomonotone vector variational inequality to be nonempty and compact.     

Some proximal algorithms for linearly constrained general variational inequalities

Based on the classical proximal point algorithm (PPA), some PPA-based numerical algorithms for general variational inequalities (GVIs) have been developed recently. Inspired by these algorithms, in this article we propose some proximal algorithms for solving linearly constrained GVIs (LCGVIs). The resulted subproblems are regularized proximally, and they are allowed to be solved either exactly or approximately.     

Modeling of EDM responses by support vector machine regression with parameters selected by particle swarm optimization

Electrical discharge machining (EDM) is inherently a stochastic process. Predicting the output of such a process with reasonable accuracy is rather difficult. Modern learning based methodologies, being capable of reading the underlying unseen effect of control factors on responses, appear to be effective in this regard. In the present work, support vector machine (SVM), one of the supervised learning methods, is applied for developing the model of EDM process. Gaussian radial basis function and ε-insensitive loss function are used as kernel function and loss function respectively. Separate models of material removal rate (MRR) and average surface roughness parameter (Ra) are developed by minimizing the mean absolute percentage error (MAPE) of training data obtained for different set of SVM parameter combinations. Particle swarm optimization (PSO) is employed for the purpose of optimizing SVM parameter combinations. Models thus developed are then tested with disjoint testing data sets. Optimum parameter settings for maximum MRR and minimum Ra are further investigated applying PSO on the developed models.     

On system of generalized vector variational inequalities

In this paper, we introduce a new system of generalized vector variational inequalities with variable preference. This extends the model of system of generalized variational inequalities due to Pang and Konnov independently as well as system of vector equilibrium problems due to Ansari, Schaible and Yao. We establish existence of solutions to the new system under weaker conditions that include a new partial diagonally convexity and a weaker notion than continuity. As applications, we derive existence results for both systems of vector variational-like inequalities and vector optimization problems with variable preference.     

Error analysis of multicategory support vector machine classifiers

The paper is related to the error analysis of Multicategory Support Vector Machine (MSVM) classifiers based on reproducing kernel Hilbert spaces. We choose the polynomial kernel as Mercer kernel and give the error estimate with De La Vallée Poussin means. We also introduce the standard estimation of sample error, and derive the explicit learning rate.