變分不等式；月行差
We obtain the existence and uniqueness of the classical solution by its equivalent parabolic quasi variational inequality.
利用其等價的抛物拟變分不等式，得到了該問題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.
利用已知的不動點定理，在非緊H 空間内得到抽象廣義變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.
從變分不等式出發，結合有限元，建立了按增量求解薄闆彈塑性彎曲問題的線性互補方程。
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.
基于摩擦約束廣義變分不等式原理，對長矩形闆镦粗進行詳細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.
第三部分讨論最高階偏導數項具有時滞的變分不等式的最優控制問題。
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.
研究了向量交通網絡平衡問題，引入了一個新的弱平衡原理，将向量交通網絡平衡問題轉化成一個變分不等式問題。
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.
基于約束集合和行駛費用函數的描述，說明了變分不等式方法在動态最優交通分配中應用的必然。
Unconventional double set parameter finite element approximation for a fourth order variational inequality with displacement obstacle is considered.
研究位移障礙下的一個四階變分不等式的非常規雙參數有限元逼近。
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.
運用該阻抗函數建立基于路徑的動态用戶最優路徑選擇變分不等式模型，給出了該模型的啟發式算法。
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.
基于摩擦約束廣義變分不等式原理，對圓柱體均勻镦粗和圓柱體非均勻镦粗等金屬成形工藝問題進行詳細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.
針對摩擦問題中具不可微泛函項的非線性混合邊界變分不等式構造了疊代分解方法，讨論了收斂性分析及誤差估計。
We establish a quasi-Newton algorithm for solving a class of variational inequality problems which subproblems are linear equations.
引言牛頓型方法是解變分不等式的一類重要數值疊代算法。
The obtained new theorem is applied to obtain an abstract variational inequality, a KKM type theorem and a fixed point theorem.
作為應用，證明了一個抽象變分不等式，一個KKM型定理和不動點定理。
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.
本文引入了幾類向量F-互補問題并給出了向量F-互補問題與廣義向量變分不等式之間的關系。
This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem.
該問題推廣了數值均衡問題，向量均衡問題和向量變分不等式問題。
As its applications, the continuity of solution mappings for a class of parametric vector optimization problem and parametric vector variational inequality is obtained.
作為應用，得到一類參數向量優化問題和參數向量變分不等式的解的連續性。
The differential equation model and the variational inequality model are equivalent for contact problems with friction and initial gaps.
有初間隙摩擦接觸問題有微分方程和變分不等式兩種等價提法。
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.
對單調變分不等式的一種新的拉格朗日方法（AL）進行讨論。
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.
另一個是通過引入變量将原邊界混合變分不等式化解為标準的凸極值問題，利用标準凸極值方法可以求解。
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.
在每步疊代計算中，新方法提出了易于計算的子問題，該子問題由強單調的線性變分不等式和良态的非線性方程系統構成。
Sensitivity analysis for multimodal equilibrium assignment variational inequality model is proposed, and a numeral example is given.
本文對多模式均衡配流變分不等式模型進行了靈敏度分析 ，并給出了數值算例。
Mathematical theory of variational inequality problems is developed from 1960's, with the studying of nonlinear problems in the continuous mechanics.
變分不等式問題的數學理論是從上世紀六十年代隨着人們對連續力學非線性問題的深入研究而發展起來的。
In this paper not to overlap domain decomposition methods for a fourth-order variational inequality problem is considered and the convergence is obtained.
本文基于一類四階變分不等式的等價形式，讨論無重疊的兩子區域分裂法，給出了方法的計算步驟，并得到了收斂性的結論。
變分不等式（Variational Inequality）是數學優化與平衡問題中的一個核心概念，尤其廣泛應用于經濟學、工程學和運籌學領域。其本質是尋找一個點，使得該點與某個集合中所有其他點的某種特定不等式關系成立。以下是詳細解釋：
變分不等式問題（VI）定義為：
給定一個閉凸集 ( K subseteq mathbb{R}^n ) 和一個映射 ( F: K to mathbb{R}^n )，目标是找到點 ( x^ in K ) 滿足：
$$ langle F(x^), x - x^ rangle geq 0, quad forall x in K $$
其中 (langle cdot, cdot rangle) 表示向量内積。該式表明，在解 ( x^ ) 處，映射 ( F ) 的方向與集合 ( K ) 中任意點 ( x ) 相對于 ( x^* ) 的偏移方向夾角為銳角或直角（來源：權威數學教材如《非線性規劃：理論與算法》）。
"變分"（Variational）的含義
源于數學中的變分法，描述函數或泛函的微小變化。在VI中體現為：解 ( x^* ) 需使 ( F ) 在 ( K ) 的局部鄰域内滿足"方向性不等式"，即解在某種意義下是穩定的。
"不等式"（Inequality）的實質
區别于方程，VI通過不等式約束描述解的魯棒性。例如在經濟學中，它可表示市場均衡條件下供需關系的非負超額收益（來源：經典論文 Facchinei & Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems）。
若 ( F ) 是某個凸函數 ( f ) 的梯度（即 ( F = abla f )），則VI等價于最小化 ( f ) 在 ( K ) 上的凸優化問題。
當 ( K ) 是非負象限時（如 ( K = mathbb{R}^n_+ )），VI退化為非線性互補問題（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：為符合原則，上述文獻均為領域内公認權威著作，具體鍊接可通過學術數據庫如SpringerLink或Google Scholar檢索獲取）
變分不等式（Variational Inequality，簡稱VI）是數學中的一個重要概念，尤其在優化、經濟學和工程學中有廣泛應用。以下是詳細解釋：
變分不等式用于描述一類尋找解的問題，其核心是在閉凸集上滿足特定不等式條件的點的存在性。具體形式為：
對于給定的閉凸集 ( X subseteq mathbb{R}^n ) 和映射 ( F: X rightarrow mathbb{R}^n )，變分不等式要求找到 ( x in X )，使得：
$$
langle F(x), y - x rangle geq 0, quad forall y in X
$$
其中，( langle cdot, cdot rangle ) 表示内積。
以簡單優化問題為例：
最小化 ( f(x) ) 受限于 ( x geq 0 )，其變分不等式形式為：
$$
x geq 0, quad
abla f(x) cdot (y - x) geq 0 quad (forall y geq 0)
$$
當 (
abla f(x) = 0 ) 時，退化為無約束優化問題。
如需更深入的理論推導或具體案例，可參考數學優化或運籌學相關文獻。
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