一类非线性拟变分不等式

本文利用伪单调算子理论研究如下变分不等式问题：求ｘ∈Ｍ，使得〈Ａｘ，ｙ－ｘ〉＋〈Ｇｘ，ｙ－ｘ〉≥〈ｆ，ｙ－ｘ〉，?ｙ∈Ｍ．并将所得结果应用于拟线性椭圆型边值问题的求解．     

一类非线性所变分不等式

本文利用伪单调算子理论研究如下变分不等式问题，求ｘ∈Ｍ，使得并将所得结果应用于拟线性椭圆型边值问题的求解。     

一类混合似变分不等式问题解的存在唯一性

本文中,我们首先给出了一类混合似变分不等式问题．接着,在Banach空间中研究了它的解的存在性和唯一性．最后,讨论了混合似变分不等式问题的扰动问题,并证明了扰动问题的解的存在唯一性定理．     

集值混合变分不等式的一类自适应惯性投影次梯度算法

本文在实Hilbert空间上引入了一类求解集值混合变分不等式新的自适应惯性投影次梯度算法.在集值映射T为f-强伪单调或单调的条件下,我们证明了由该自适应惯性投影次梯度算法所产生的序列强收敛于集值混合变分不等式问题的的唯一解.     

集值映射的变分不等式和补问题及应用

The purpose of this paper is to study the variational inequality and complementarity problem for set-valued mapping in Banach spaces.The results presented in this paper generalize and improve some recent results on finite dimensional spaces,Hiblert spaces,single-valued mapping.As applications.we utilize our results to study saddle point problems and two programming problems in infinite dimensional spaces.     

一类非线性拟变分不等式及其应用

本义研究了如下变分不等式问题：求ｕ∈Ｍ使得ａ（ｕ，ｖ－ｕ）＋〈Ｇｕ，ｖ－ｕ〉≥（ｆ，ｖ－ｕ），(?)ｕ∈Ｍ．并将所得结论应用于二阶半线性椭圆型边值问题的求解．     

广义集值拟变分不等式

本文研究了一类广义集值拟变分不等式,得到了这类广义集值拟变分不等式解存在的一个充分条件,并给出了近似解的迭代算法.     

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．     

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。     

非单调变分不等式黄金分割算法研究

该文考虑变分不等式的梯度投影算法,给出了一种非单调变分不等式的黄金分割算法,所给出的算法特点结合了惯性加速方法,无需知道映射的Lipschitz常数,且步长是非单调递减的.在一定的条件下,算法的收敛性被证明.最后给出数值实验结果.     

一类集值非线性混合变分包含问题的逼近解

在Hilbert空间中讨论了一类集值非线性混合变分包含问题逼近解的存在性,建立了变分包含问题与其预解方程的等价性,获得了3个迭代算法并研究了算法的收敛性．该结果推广统一了近期一些学者关于变分包含问题的相关结果．     

集值非线性混合变分包含问题的一类新的迭代算法

在Hilbert空间中讨论一类广义集值非线性混合变分包含问题近似解的存在性，建立变分包含与广义预解方程的等价性，形成了迭代算法并研究了算法的收敛性．     

Optimal Control of Nonlinear Transport-Flow Mixed Variational Problems

Optimal control of nonlinear transport-flow mixed variational problems are studied qualitatively, and the solvability analysis of the transport and flow mixed state systems is performed on the basis of primal and dual evolution duality principles. Corresponding primal and dual mixed optimality conditions are established by the application of some fundamental perturbation conjugate duality results recently proposed [8 G. Alduncin (2013). Optimal control of evolution mixed variational inclusions. Applied Mathematics and Optimization 68:445–473.[Crossref], [Web of Science ®] , [Google Scholar]]. Further, for computational purposes, two- and three-field proximation penalty-duality algorithms in the resolution of the mixed optimality conditions are finally presented and discussed.     

关于变分不等式与不动点问题(英文)

In this paper,A strong convergence theorem for a finite family of nonexpansive mappings and relaxed cocoercive mappings based on an iterative method in the framework of Hilbert spaces is established.     

Unconstrained Methods for Generalized Nonlinear Complementarity and Variational Inequality Problems

In this paper, we construct unconstrained methods for the generalized nonlinear complementarity problem and variational inequalities. Properties of the correspondent unconstrained optimization problem are studied. We apply these methods to the subproblems in trust region method, and study their interrelationships. Numerical results are also presented.     

Solvability of Variational Inequality Problems

This paper presents the new concept of exceptional family of elements for the variational inequality problem with a continuous function over a general unbounded closed convex set. We establish a characterization theorem that can be used to derive several new existence and compactness conditions on the solution set. Our findings generalize well-known results for various types of variational inequality problems. For a pseudomonotone variational inequality problem, our new existence conditions are both sufficient and necessary.     

New Alternating Direction Method for a Class of Nonlinear Variational Inequality Problems

The alternating direction method is an attractive method for a class of variational inequality problems if the subproblems can be solved efficiently. However, solving the subproblems exactly is expensive even when the subproblem is strongly monotone or linear. To overcome this disadvantage, this paper develops a new alternating direction method for cocoercive nonlinear variational inequality problems. To illustrate the performance of this approach, we implement it for traffic assignment problems with fixed demand and for large-scale spatial price equilibrium problems.     

Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems

This paper considers the expected residual minimization (ERM) method proposed by Luo and Lin (J. Optim. Theory Appl. 140:103–116, 2009) for a class of stochastic variational inequality problems. Different from the work mentioned above, the function involved is assumed to be nonlinear in this paper. We first consider a quasi-Monte Carlo method for the case where the underlying sample space is compact and show that the ERM method is convergent under very mild conditions. Then, we suggest a compact approximation approach for the case where the sample space is noncompact. This work was supported in part by Project 10771025 supported by NSFC and SRFDP 20070141063 of China.     

摩擦问题中的边界混合变分不等式

本文以弹性力学中的摩擦问题为背景,讨论了非线性、不可微的混合变分不等式解的存在唯一性,给出相应的边界变分不等式及其解的存在唯一性。为使用边界元方法数值求解提供了理论依据。