变分不等式及其应用 摘要 变分不等式是一类重要的非线性问题，它在工程、经济、控制理论等领域广泛应用。变分不等式问题的数学理论最开始应用于解决均衡问题，在此模型中，函数来自对应势能的一阶变分，因此而得名．作为经典变分问题的推广和发展，变分不等式的形式也更多样化。本文主要研究变分不等式的由来，变分不等式的导出以及一些变分不等式的应用． 第一章为预备知识，主要介绍了凸泛函、上下半连续泛函、次连续、Ferchet微分和单调映像等的一些定义，为下文更好的引出变分不等式的概念、导出和应用提供了理论依据。 第二章具体的提出变分不等式的概念并给出一些变分不等式的常见例子。 第三章主要通过可微函数的极值问题、不可微函数的极值问题、Hilbert 空间的投影问题、分布参数系统控制问题等一些问题的探讨说明导出变分不等式一些方法。 第四章研究一类非线性拟变分不等式并应用于二阶半线性椭圆型边值问题。 关键词：变分不等式，极值问题，椭圆方程，边值问题
VARIATIONAL INEQUALITY AND ITS APPLICATION ABSTRACT Variational inequalities are important nonlinear problems, it has been widely applied in the fields of engineering, economics, control theory. The mathematical theory of variational inequality problem is originally applied to solve equilibrium problem. In this model, the function comes from the first-order variation of the corresponding potential energy, so it is called variational inequality problem. As the generalization and development of classical variational problems, the form of variational inequalities should be diversification. In this paper, i study the origin, derivation, and applications of variational inequalities. The first chapter is is Preliminaries. In this chaper, i list the definitions of convex functional, upper and lower semi-continuous functional, consecutive, Ferchet differential, montonous map, and so on. They are used forunderstanding the concept, derivation, and applications of variational inequality. In the second chapter, i introduce the concept of variational inequalities and give some common examples of variational inequalities. In the third chapter, by consdering differentiable functions’ extremum problems, non-differentiable functions’ extremum problems, the projection in Hilbert space, control systems of distributed parameter and some other issues, i study the methods of variational inequalities’ derivation. In the fourth chapter, a class of nonlinear quasi-variational inequalitie is introduce, and it is applied to solve second order semi-linear elliptic boundary value problems. Key words:Variational inequalities, extremum problem, elliptic equation，boundary value problem
求解变分不等式： 例2：.]5,5[,0)(,0),(**n w w g v w v F -∈?≤≥-（n 可以是维数，在我们计算的过程中，可以取100,200,……1000维） ???? ? ??++=n v n v e v e v v F 11)(，w Aw w g ,)(=，A 是一个n n ?对称矩阵，可随机生成。 例 1：.]5,5[,0)(,0),(**n w w g v w v F -∈?≤≥- 其中?????? ? ??---+??????? ????????? ??---=34680211220210421224)(4321v v v v x F ,w w w g ,)(-=. 其解为（4/3,7/9,4/9,2/9）。(不变) 用迭代序列编程求解： 高维迭代 clc; k=0; k_inner=1000; time0=cputime; n=4; v0=0*rand(n,1); p0=1; Q=eye(n); b=5*diag(Q);%盒子的上界 a=-5*diag(Q)%盒子的下届 mu=0.03;
%F函数的输入如下 F=zeros(n,1) for i=1:n F(i,1)=v0(i)+exp(v0(i)); end barp0=max(0,p0-mu*sum(v0.^2)); % barv0=zeros(4,1); for i=1:n if v0(i)-mu*(F(i)+barp0*v0(i))<=a(i) barv0(i)=a(i); elseif v0(i)-mu*(F(i)+barp0*v0(i))>=b(i) barv0(i)=b(i); else barv0(i)=v0(i)-mu*(F(i)+barp0*v0(i)); end end