## An Approximation Theorem of Variational Inequalities Under Bounded Rationality

Qiu Xiaoling, Jia Wensheng,

 基金资助: 国家自然科学基金.  11561013人社部留学归国人员择优资助项目.  2015192贵州省科技厅联合基金.  QKH[2014]7643贵州省科技厅联合基金.  [2016]7425贵州大学引进人才基金.  201405贵州大学引进人才基金.  201811

 Fund supported: the NSFC.  11561013the Project for the Selection of Returnees from Overseas Studies by the Ministry of Human Resources and Social Sciences.  2015192the Science and Technology Foundation of Guizhou Province.  QKH[2014]7643the Science and Technology Foundation of Guizhou Province.  [2016]7425the Introduced Talent Foundation of Guizhou University.  201405the Introduced Talent Foundation of Guizhou University.  201811

Abstract

In this paper, basing on Simon's bounded rationality theory, we first prove and construct an approximation theorem for variational inequalities problems, which provide theoretical support for many relevant different algorithms. Simon's bounded rationality is illustrated and bounded rationality is approaching to full rationality as its ultimate goal. Then, by the methods of set-valued analysis, bounded rationality approximation theory is used for the convergence analysis of solutions of variational inequalities problems. In the sense of Baire category, we obtain the generic convergence of the solutions of monotone variational inequalities problems, in both cases that the function disturbance and the function and constraint set disturbance.

Keywords： Bounded rationality ; Monotone variational inequalities ; Approximation theorem ; Set-valued mapping ; Generic convergence

Qiu Xiaoling, Jia Wensheng. An Approximation Theorem of Variational Inequalities Under Bounded Rationality. Acta Mathematica Scientia[J], 2019, 39(4): 730-737 doi:

## 1 引言

$X$是Hilbert空间$E$中的非空子集, $E$中的内积和范数分别记为$\langle\cdot, \cdot\rangle $$\|\cdot\|,$$ T:X\to E$是一个函数,由$T$决定的变分不等式问题是

## 2 预备知识

(1)如果对$X$中的任意开集$G, G\supset F(x)$ (或$G\cap F(x)\neq {\emptyset}$),存在$x$的开邻域$O(x)$,使得$\forall x'\in O(x), $$G\supset F(x') ( G\cap F(x')\neq {\emptyset} ),则称集值映射 F$$ x$上是上半连续的(或下半连续的).

(2)如果$F $$x 上既上半连续的又下半连续,则称集值映射 F$$ x$上是连续的.

(3)如果$\forall x\in X$,集值映射$F $$x 上是连续的(或上半连续的,或下半连续的),则 F$$ X$上是连续的(或上半连续的,或下半连续的).

(4)称$F $$X 上是usco映射,如果 F$$ X$是上半连续的,且$\forall x\in X$, $F(x)$是非空紧集.

(5)称集值映射$F$是闭的,如果$GraphF = \{(x, y)\in X\times Y: y\in F(x)\} $$X\times Y 中的闭集. (6)如果 F$$ X$上是usco映射,则$F$是闭映射.

(7)如果$F$是闭的, $Y$为紧空间,则集值映射$F $$X 上是上半连续的. 集值映射的上半连续与下半连续是两个不同的概念,一般不能从一个成立推出另一个成立,但Fort[15]指出了它们之间的某种联系,其在通有连续性研究中具有至关重要的作用. Fort引理[15] 设 X 是一个Hausdorff拓扑空间, Y 是一个度量空间, F:X\rightarrow P_{0}(Y) 是一个usco映射,则存在 X 中的一个剩余集 Q ,使得 \forall x\in Q , F$$ x$是下半连续的,从而是连续的.

(ⅰ)对每个$n = 1, 2, \cdots,$函数$T^n:X\to E$满足$\sup\limits_{x\in X}\|T^n(x)-T(x)\|\to 0\ (n\to\infty)$,其中$T:X\to\mathbb{R}$是连续的;

(ⅱ)对每个$n = 1, 2, \cdots, $$A_n$$ X$中的子集,且$h(A_n, A)\to 0 \ (n\to\infty)$,其中$A $$X 中的非空紧集; (ⅲ)对每个 n = 1, 2, \cdots,$$ x_n\in X, $$d(x_n, A_n)\to 0 ,满足 其中 \epsilon_n >0$$ \epsilon_n\to 0 \ (n\to \infty).$

(1)存在$\{x_n\}$的一个收敛子列$\{x_{n_k}\}$,即$x_{n_k}\to x^* (n\to\infty) $$x^*\in A ; (2)对任意的 y\in A , \langle T(x^*), y-x^*\rangle\geq 0; (3)若变分不等式 T 的解集是单点集,必有 x_n\to x^*. (1)由于 d(x_n, A_n)\to 0,$$ \exists x'_n\in A_n$,使得$d(x_n, x'_n)\to 0$.再由$h(A_n, A)\to 0\ (n\to\infty)$,根据引理2.4,结论显然成立.

(2)根据(1)的结果,我们不妨设$x_n\to x^*$.反证.如果结论不正确,那么存在点$y_0\in A$使得$\langle T(x^*), y_0-x^*\rangle<0;$

(ⅰ)对每个$n = 1, 2, \cdots,$函数$T^n:X\to E$满足$\sup\limits_{x\in X}| T^n(x)-T(x)| \to 0\ (n\to\infty)$,其中$T:X\to\mathbb{R}$是连续的;

(ⅱ)对每个$n = 1, 2, \cdots, $$A_n$$ X$中的子集,且$h(A_n, A)\to 0 \ (n\to\infty)$,其中$A $$X 中的非空紧集; (ⅲ)对每个 n = 1, 2, \cdots,$$ x_n\in A_n,$满足

(2)对任意的$y\in A$, $\langle T(x^*), y-x^*\rangle\geq 0;$

(3)若变分不等式$T$的解集是单点集,必有$x_n\to x^*.$

(ⅰ)对每个$n = 1, 2, \cdots,$函数$T^n:X\to E$满足$\sup\limits_{x\in X}| T^n(x)-T(x)| \to 0\ (n\to\infty)$,其中$T:X\to \mathbb{R}$是连续的;

(ⅱ) $A $$X 中的非空紧集; (ⅲ)对每个 n = 1, 2, \cdots,$$ x_n\in A $$A$$ \epsilon_n$的近似解,即满足

(2)对任意的$y\in A$, $\langle T(x^*), y-x^*\rangle\geq 0;$

(3)若变分不等式$T$的解集是单点集,必有$x_n\to x^*.$

## 4 变分不等式的通有收敛性

$\forall T\in M$,选取函数序列$\{T^n\}$ (对$T^n$没有任何连续性要求),使$\sup\limits_{x\in X}\| T^n(x)-T(x)\| \to 0,$选取$x_n\in X,$使$\langle T^n(x_n), y-x_n\rangle\geq -\epsilon_n$对任意的$y\in X$都成立,其中$\epsilon_n >0 $$\epsilon_n\to 0$$ (n\to \infty).$

由文献[13,定理7.4.4],存在$M$中的稠密剩余集$Q_1$,使得$\forall T\in Q_1$, $T$的解是单点集,再由定理3.1中的结论(3),必有$x_n\to x.$

$X$是Hilbert空间$E$中的非空凸子集(不必紧),问题空间$M_1$构造如下

$K(X)$表示$X$中所有非空凸紧子集组成的集合且装备了Hausdorff距离$h$.$E$的完备性, ($K(X), h$)也是完备度量空间,见文献[21,定理3.85].

$S(u)$表示$T $$A 中的所有解.显然 S(u)\not =\varnothing ,且映射 u\to S(u) 定义了一个集值映射 S:M_1\to P_0(X) .由文献[7,引理4.3],得以下引理. 引理4.2 S:M_1\to P_0(X) 是一个usco映射. 再根据文献[7,定理4.1],有以下定理. 定理4.2 存在 M_1 中的一个稠密剩余集 Q_2 ,使得 Q_2 中的每个 u = (T, A),$$ S(u)$是单点集.

$\forall u = (T, A)\in M_1$,选取函数序列$\{T^n\}$ (对$T^n$没有任何连续性要求),使$\sup\limits_{x\in X}\| T^n(x)-T(x)\| \to 0,$选取$X$中的子集序列$\{A_n\}$ (对$A_n$没有任何紧性或闭性要求),使$h(A_n, A)\to 0 $$(n\to\infty); 选取 x_n\in A_n, 使 \langle T^n(x_n), y-x_n\rangle\geq -\epsilon_n 对任意的 y\in A_n 都成立,其中 \epsilon_n >0$$ \epsilon_n\to 0 \ (n\to \infty).$

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