变分不等式；月行差
We obtain the existence and uniqueness of the classical solution by its equivalent parabolic quasi variational inequality.
利用其等价的抛物拟变分不等式，得到了该问题古典解的存在唯一性。
By using a known fixed point theorem, an existence theorem of solutions of generalized variational inequality is proved in noncompact H space.
利用已知的不动点定理，在非紧H 空间内得到抽象广义变分不等式解的存在性定理；
Finite Elemnent-Linear Complementary Equation is established for analysis of elastoplastic thin plate-bending problems based on variational inequality principle.
从变分不等式出发，结合有限元，建立了按增量求解薄板弹塑性弯曲问题的线性互补方程。
The plastic process analyses on rectangular plate upsetting were made in detail on the basis of the principle of generalized variational inequality subjected to friction.
基于摩擦约束广义变分不等式原理，对长矩形板镦粗进行详细的塑性加工工步分析。
Finally in the third part, we study the optimal control problems for the variational inequality with delays in the highest order spatial derivatives.
第三部分讨论最高阶偏导数项具有时滞的变分不等式的最优控制问题。
In this paper we research a vector equilibrium problem of traffic network problems. We introduce a new weak equilibrium principle and transfer the problem to a variational inequality problem.
研究了向量交通网络平衡问题，引入了一个新的弱平衡原理，将向量交通网络平衡问题转化成一个变分不等式问题。
The necessity of the application of variational inequality method to dynamic traffic assignment is explained based on the description of the set of constraints and travel cost function.
基于约束集合和行驶费用函数的描述，说明了变分不等式方法在动态最优交通分配中应用的必然。
Unconventional double set parameter finite element approximation for a fourth order variational inequality with displacement obstacle is considered.
研究位移障碍下的一个四阶变分不等式的非常规双参数有限元逼近。
A route-based variational inequality model and a heuristic algorithm of the dynamic user optimal route choice problem were proposed by the link travel time function.
运用该阻抗函数建立基于路径的动态用户最优路径选择变分不等式模型，给出了该模型的启发式算法。
The plastic process analyses on uniform and non uniform cylindrical upsetting were made in detail on the basis of the principle of generalized variational inequality with friction.
基于摩擦约束广义变分不等式原理，对圆柱体均匀镦粗和圆柱体非均匀镦粗等金属成形工艺问题进行详细的塑性加工工步分析。
Based on the boundary mixed variational inequality with non-differentiable functional in friction problem, the iterative decomposition method and its convergence analysis are presented.
针对摩擦问题中具不可微泛函项的非线性混合边界变分不等式构造了迭代分解方法，讨论了收敛性分析及误差估计。
We establish a quasi-Newton algorithm for solving a class of variational inequality problems which subproblems are linear equations.
引言牛顿型方法是解变分不等式的一类重要数值迭代算法。
The obtained new theorem is applied to obtain an abstract variational inequality, a KKM type theorem and a fixed point theorem.
作为应用，证明了一个抽象变分不等式，一个KKM型定理和不动点定理。
In this paper, we introduce several classes of vector F-complementarity problems and give relations between vector F-complementarity problems and general vector variational inequality problems.
本文引入了几类向量F-互补问题并给出了向量F-互补问题与广义向量变分不等式之间的关系。
This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem.
该问题推广了数值均衡问题，向量均衡问题和向量变分不等式问题。
As its applications, the continuity of solution mappings for a class of parametric vector optimization problem and parametric vector variational inequality is obtained.
作为应用，得到一类参数向量优化问题和参数向量变分不等式的解的连续性。
The differential equation model and the variational inequality model are equivalent for contact problems with friction and initial gaps.
有初间隙摩擦接触问题有微分方程和变分不等式两种等价提法。
This paper addresses a new augmented Lagrange method(AL) for monotone variational inequality(VI), which needs only to solve one stongly monotone sub-VI problem.
对单调变分不等式的一种新的拉格朗日方法（AL）进行讨论。
Second, introducing the transformation, the mixed variational inequality is reduced to a standard convex optimization problem, it can be solved by any standard methods of convex optimization.
另一个是通过引入变量将原边界混合变分不等式化解为标准的凸极值问题，利用标准凸极值方法可以求解。
At each iteration, the proposed subproblem consists of a strongly monotonic linear variational inequality and a well-conditioned system of nonlinear equations, which is easily to be solved.
在每步迭代计算中，新方法提出了易于计算的子问题，该子问题由强单调的线性变分不等式和良态的非线性方程系统构成。
Sensitivity analysis for multimodal equilibrium assignment variational inequality model is proposed, and a numeral example is given.
本文对多模式均衡配流变分不等式模型进行了灵敏度分析 ，并给出了数值算例。
Mathematical theory of variational inequality problems is developed from 1960's, with the studying of nonlinear problems in the continuous mechanics.
变分不等式问题的数学理论是从上世纪六十年代随着人们对连续力学非线性问题的深入研究而发展起来的。
In this paper not to overlap domain decomposition methods for a fourth-order variational inequality problem is considered and the convergence is obtained.
本文基于一类四阶变分不等式的等价形式，讨论无重叠的两子区域分裂法，给出了方法的计算步骤，并得到了收敛性的结论。
变分不等式（Variational Inequality）是数学优化与平衡问题中的一个核心概念，尤其广泛应用于经济学、工程学和运筹学领域。其本质是寻找一个点，使得该点与某个集合中所有其他点的某种特定不等式关系成立。以下是详细解释：
变分不等式问题（VI）定义为：
给定一个闭凸集 ( K subseteq mathbb{R}^n ) 和一个映射 ( F: K to mathbb{R}^n )，目标是找到点 ( x^ in K ) 满足：
$$ langle F(x^), x - x^ rangle geq 0, quad forall x in K $$
其中 (langle cdot, cdot rangle) 表示向量内积。该式表明，在解 ( x^ ) 处，映射 ( F ) 的方向与集合 ( K ) 中任意点 ( x ) 相对于 ( x^* ) 的偏移方向夹角为锐角或直角（来源：权威数学教材如《非线性规划：理论与算法》）。
"变分"（Variational）的含义
源于数学中的变分法，描述函数或泛函的微小变化。在VI中体现为：解 ( x^* ) 需使 ( F ) 在 ( K ) 的局部邻域内满足"方向性不等式"，即解在某种意义下是稳定的。
"不等式"（Inequality）的实质
区别于方程，VI通过不等式约束描述解的鲁棒性。例如在经济学中，它可表示市场均衡条件下供需关系的非负超额收益（来源：经典论文 Facchinei & Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems）。
若 ( F ) 是某个凸函数 ( f ) 的梯度（即 ( F = abla f )），则VI等价于最小化 ( f ) 在 ( K ) 上的凸优化问题。
当 ( K ) 是非负象限时（如 ( K = mathbb{R}^n_+ )），VI退化为非线性互补问题（NCP）：
$$ x^ geq 0, quad F(x^) geq 0, quad langle x^, F(x^) rangle = 0 $$
Kinderlehrer, D., & Stampacchia, G. (1980). An Introduction to Variational Inequalities and Their Applications. Academic Press.
Facchinei, F., & Pang, J. S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer.
Nagurney, A. (1999). Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers.
（注：为符合原则，上述文献均为领域内公认权威著作，具体链接可通过学术数据库如SpringerLink或Google Scholar检索获取）
变分不等式（Variational Inequality，简称VI）是数学中的一个重要概念，尤其在优化、经济学和工程学中有广泛应用。以下是详细解释：
变分不等式用于描述一类寻找解的问题，其核心是在闭凸集上满足特定不等式条件的点的存在性。具体形式为：
对于给定的闭凸集 ( X subseteq mathbb{R}^n ) 和映射 ( F: X rightarrow mathbb{R}^n )，变分不等式要求找到 ( x in X )，使得：
$$
langle F(x), y - x rangle geq 0, quad forall y in X
$$
其中，( langle cdot, cdot rangle ) 表示内积。
以简单优化问题为例：
最小化 ( f(x) ) 受限于 ( x geq 0 )，其变分不等式形式为：
$$
x geq 0, quad
abla f(x) cdot (y - x) geq 0 quad (forall y geq 0)
$$
当 (
abla f(x) = 0 ) 时，退化为无约束优化问题。
如需更深入的理论推导或具体案例，可参考数学优化或运筹学相关文献。
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