西南大学学报 (自然科学版)  2017, Vol. 39 Issue (8): 101-107.  DOI: 10.13718/j.cnki.xdzk.2017.08.015
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  • 一类锥约束变分不等式问题的间隙函数和误差界    [PDF全文]
    董文1, 欧小庆2, 李金富1, 陈加伟1     
    1. 西南大学 数学与统计学院,重庆 400715;
    2. 重庆人文科技学院 管理学院,重庆 401524
    摘要:鉴于间隙函数与误差界在优化方法中有重要的作用,特别地,误差界能刻画可行点和变分不等式解集之间的有效估计距离.利用像空间分析法,构造了带锥约束变分不等式的间隙函数.然后,利用此间隙函数,得到了带锥约束变分不等式的误差界.
    关键词约束变分不等式    像空间分析    间隙函数    误差界    

    众所周知,变分不等式在优化理论和方法、经济管理与交通等方面都有着广泛的应用[1-3].间隙函数的概念首次被引入用于凸优化问题的研究,随后才应用于变分不等式.一方面,由于间隙函数将变分不等式问题转换为等价的优化问题,故可用优化求解法和算法来求得变分不等式的解.另一方面,间隙函数在设计新的全局收敛算法和分析一些迭代方法的收敛速率以及导出误差界等方面非常有用[4-5].

    本文主要研究带锥约束的变分不等式,旨在利用像空间分析得到间隙函数.像空间分析法是一个非常有力的工具,用于研究各种类型的问题,它把各类问题等价地表示成一个参数系统的不可行性以及约束优化像空间中两个集合的分离性,近年来,像空间分析法受到相当大的关注[6-9].

    本文由三部分组成.第一部分简要回顾了一些准备知识,并分析了像空间分析的一般特征;第二部分利用像空间分析,给出了带锥约束变分不等式的两个间隙函数;第三部分利用两个间隙函数,得到了在逆强伪单调假设条件下带锥约束变分不等式的解集的误差界.

    1 预备知识

    首先回顾一些符号和定义,集合$M\subseteq {{\mathbb{R}}^{n}}$的内部和边界分别表示为intM∂M.设$ K\subset {{\mathbb{R}}^{n}}$为内部非空的闭凸点锥.

    给定函数$f:{{\mathbb{R}}^{n}}\to {{\mathbb{R}}^{n}}$.本文考虑如下带锥约束的变分不等式:找到${{x}^{*}}\in {{\mathbb{R}}^{n}}$,使得

    $ f\left( {{x^ * }} \right) \in \Omega \;\;\;\;\;\;\;{\left( {y - f\left( {{x^ * }} \right)} \right)^{\rm{T}}}{x^ * } \ge 0\;\;\;\;\;\;\;\forall y \in \Omega $ (1)

    其中$\Omega =\left\{ y\in {{\mathbb{R}}^{n}}:g\left( y \right)\in D \right\}, g:{{\mathbb{R}}^{n}}\to {{\mathbb{R}}^{m}}$为向量值映射,$ D\subseteq {{\mathbb{R}}^{m}}$为内部非空的闭凸点锥.

    接下来,我们给出变分不等式(1) 像的主要特点.给定${{x}^{*}}\in {{\mathbb{R}}^{n}}$,定义映射

    $ {A_{{x^ * }}}:{\mathbb{R}^n} \to {\mathbb{R}^{1 + 2m}}\;\;\;\;\;{A_{{x^ * }}}\left( y \right) = \left( {{{\left( {f\left( {{x^ * }} \right) - y} \right)}^{\text{T}}}{x^ * }, g\left( y \right), g\left( {f\left( {{x^ * }} \right)} \right)} \right) $

    考虑集合

    $ \begin{array}{*{20}{c}} {\mathscr{K}\left( {{x^ * }} \right) = \left\{ {\left( {u, v, \tau } \right) \in {\mathbb{R}^{1 + 2m}}:\left( {u, v, \tau } \right) = {A_{{x^ * }}}\left( y \right), y \in {\mathbb{R}^n}} \right\}} \\ {\mathscr{H} = \left\{ {\left( {u, v, \tau } \right) \in {\mathbb{R}^{1 + 2m}}:u > 0, \left( {v, \tau } \right) \in D \times D} \right\}} \end{array} $

    其中$\mathscr{K}\left( {{x}^{*}} \right)$称为变分不等式(1) 的像,$ {{\mathbb{R}}^{1+2m}}$称为像空间.显然,${{x}^{*}}\in {{\mathbb{R}}^{n}}$是变分不等式(1) 的解,当且仅当广义系统

    $ {A_{{x^ * }}} \in \mathscr{H}\;\;\;\;y \in {\mathbb{R}^n} $ (2)

    是不可行的,或等价于

    $ \mathscr{K}\left( {{x^ * }} \right) \cap \mathscr{H} = Ø $

    对于定义在集合$X\subseteq {{\mathbb{R}}^{n}} $$\alpha \in \mathbb{R} $的函数h,集合

    $ {\rm{le}}{{\rm{v}}_{ \ge \alpha }}h = \left\{ {x \in X:h\left( x \right) \ge \alpha } \right\}\;\;\;\;\;{\rm{lev}} \ge {}_\alpha h = \left\{ {x \in X:h\left( x \right) > \alpha } \right\} $

    分别称为函数h的非负水平集和正水平集.

    定义1  给定e∈-intK,定义Gerstewitz函数${{\xi }_{e, K}}:{{\mathbb{R}}^{n}}\to \mathbb{R}$为:

    $ {\xi _{e, K}}\left( y \right) = \min \left\{ {r \in \mathbb{R}:y \in re + K} \right\}\;\;\;\;\;\;y \in {\mathbb{R}^n} $

    命题1[11-12]  对任意给定的e∈-intK$y\in {{\mathbb{R}}^{n}}$$r\in \mathbb{R}$,有下面的结论成立:

    (ⅰ) ξeK(y)<ryre+intK

    (ⅱ) ξeK(y)≤ryre+K

    (ⅲ) ξeK(y)=ryre+∂K

    (ⅳ) Gerstewitz函数ξeK${{\mathbb{R}}^{n}}$上是下降的,即

    $ x - y \in K \Rightarrow {\xi _{e, K}}\left( x \right) \leqslant {\xi _{e, K}}\left( y \right)\;\;\;\;\;\;\;\;\forall x, y \in {\mathbb{R}^n} $
    2 间隙函数

    本节构造了变分不等式(1) 的两个间隙函数.首先回顾一下变分不等式的间隙函数的基本定义.

    定义2  称函数$P:{{\mathbb{R}}^{n}}\to \mathbb{R}\cup \left\{ +\infty \right\}$为变分不等式(1) 的间隙函数,若

    (ⅰ) $P\left( x \right)\ge 0, \forall x\in {{\mathbb{R}}^{n}}$

    (ⅱ) P(x*)=0当且仅当x*S.

    θ>0.考虑函数

    $ \begin{array}{*{15}{c}} {{P_1}\left( x \right) = \mathop {\sup }\limits_{\left( {{\text{u}}, {\text{v}}, {\tau }} \right) \in \mathscr{K}\left( x \right)} \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta u-\lambda {\xi _{e, D}}\left( v \right)-\beta {\xi _{e, D}}\left( \tau \right)} \right]\;\;\;\;\;\;\;\;\;\forall x \in {\mathbb{R}^n}} \\ {{P_2}\left( x \right) = \mathop {\sup }\limits_{\left( {{\text{u}}, {\text{v}}, {\tau }} \right) \in \mathscr{K}\left( x \right)} \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }}\\ \left[{\bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right]\;\;\;\;\;\;\;\forall x \in {\mathbb{R}^n}} \end{array} $

    其中e∈-intDr为正实数,扩张函数$\sigma :{{\mathbb{R}}^{m}}\to \mathbb{R} $上半连续且

    $ \arg \mathop {\min }\limits_{x \in {\mathbb{R}^m}} \sigma \left( z \right) = \left\{ {{0_{{\mathbb{R}^m}}}} \right\}\;\;\;\;\;\;\;\;\;\;\sigma \left\{ {{0_{{\mathbb{R}^m}}}} \right\} = 0 $

    引理1  设θ>0,且

    $ \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) = \bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\} - D} \left( { - \lambda {\xi _{e, D}}\left( z \right) - r\sigma \left( z \right)} \right) +\\ \mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)\;\;\;\;\;\;\;\lambda, \beta \in {\mathbb{R}_ + } $

    $ \mathscr{H} = \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} = {\text{le}}{{\text{v}}_{ > 0}}\mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right) $ (3)

      了得到式(3),只需证

    $ \mathscr{H} = \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} \supseteq {\text{le}}{{\text{v}}_{ > 0}}\mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right) \supseteq \mathscr{H} $ (4)

    首先证式(4) 中的等号成立.对任意$\lambda, \beta \in {{\mathbb{R}}_{+}}, \left( u, v, \tau \right)\in \mathscr{H}$,均有θu>0.又由命题1(ⅱ),${{0}_{{{\mathbb{R}}_{m}}}}\in \left( \left\{ v \right\}-D \right)\cap \left( \left\{ \tau \right\}-D \right)$$ \sigma \left( {{0}_{{{\mathbb{R}}_{m}}}} \right)=0$

    $ \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) = \bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\} - D} \left( { - \lambda {\xi _{e, D}}\left( z \right) - r\sigma \left( z \right)} \right) +\\ \mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right) > 0 $

    这意味着

    $ \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} \supseteq \mathscr{H} $ (5)

    下证

    $ \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} \subseteq \mathscr{H} $ (6)

    为此,只需证当$\left( u, v, \tau \right)\notin \mathscr{H}$时,存在$\lambda, \beta \in {{\mathbb{R}}_{+}} $,使得ω(uvτθλβ)≤0即可.设$\left( u, v, \tau \right)\notin \mathscr{H}$,下面分3种情形来讨论:

    情形1  若u≤0,(vτ)∈D×D,则不妨设λβ=0.由于$\underset{z\in {{\mathbb{R}}^{m}}}{\mathop{\arg \min }}\, \sigma \left( z \right)=\left\{ {{0}_{{{\mathbb{R}}^{m}}}} \right\}, \sigma \left\{ {{0}_{{{\mathbb{R}}^{m}}}} \right\}=0$,因此σ(z)≥0对所有$z\in {{\mathbb{R}}^{m}} $均成立.再由${{0}_{{{\mathbb{R}}^{m}}}}\in \left( \left\{ v \right\}-D \right)\cap \left( \left\{ \tau \right\}-D \right)$$\sigma \left( {{0}_{{{\mathbb{R}}_{m}}}} \right)=0 $

    $ \mathop {\sup }\limits_{z \in \left\{ v \right\} - D} \left( { - r\sigma \left( z \right)} \right) = 0\;\;\;\;\;\;\mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - r\sigma \left( t \right)} \right) = 0 $

    故有ω(uvτθλβ)=θu≤0.

    情形2  若u>0,vDτD,则不妨设$\lambda =\frac{{{{\bar{\theta }}}{u}}}{{{\xi }_{e, D}}\left( v \right)}, \beta =0$,由命题1(ⅱ)知$\lambda, \beta \in {{\mathbb{R}}_{+}} $.再由ξeD(·)在$\mathbb{R}^{m}$上是下降的且对所有$z\in {\mathbb{R}^{m}},\sigma \left( z \right)\ge 0$.从而有

    $ \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) \leqslant \bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\} - D} \left( { - \lambda {\xi _{e, D}}\left( z \right)} \right) \leqslant \bar \theta u - \lambda {\xi _{e, D}}\left( v \right) = 0 $

    情形3  若u>0,vDτD,则不妨设λ=0,$\beta =\frac{\bar{\theta }u}{{{\xi }_{e, D}}\left( \tau \right)}$,由命题1(ⅱ)知$\lambda, \beta \in {{\mathbb{R}}_{+}} $.再由ξeD(·)在${{\mathbb{R}}^{m}}$上是下降的且对所有$z\in {\mathbb{R}^{m}},\sigma \left( z \right)\ge 0$.故有

    $ \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) \leqslant \bar \theta u - \beta {\xi _{e, D}}\left( v \right) = 0 $

    由式(5) 和式(6) 可得式(4) 中的等号成立.

    接下来,我们证式(4) 的第一个包含关系.任取$ \left( u, v, \tau \right)\in \text{le}{{\text{v}}_{>0}}\underset{\lambda, \beta \in {{\mathbb{R}}_{+}}}{\mathop{\text{lnf}}}\, \omega \left( \cdot \ ;\bar{\theta }, \lambda, \beta \right)$,均有

    $ \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) > 0 $

    $ \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) > 0\;\;\;\;\;\;\;\forall \lambda, \beta \in {\mathbb{R}_ + } $

    从而

    $ \left( {u, v, \tau } \right) \in \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} $

    这意味着

    $ \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} \supseteq {\text{le}}{{\text{v}}_{ > 0}}\bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {\omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right)} $

    下证式(4) 的第二个包含关系.任取$\left( u, v, \tau \right)\in \mathscr{H}$,由命题1(ⅱ)和$ {{0}_{{{\mathbb{R}}_{m}}}}\in \left( \left\{ v \right\}-D \right)\cap \left( \left\{ \tau \right\}-D \right)$

    $ \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \omega \left( {u, v, \tau ;\bar \theta, \lambda, \beta } \right) \geqslant \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta u-\left( {\lambda + \beta } \right){\xi _{e, D}}\left( {{0_{{\mathbb{R}^m}}}} \right)-2r\sigma \left( {{0_{{\mathbb{R}^m}}}} \right)} \right] = \bar \theta u > 0 $

    于是有

    $ {\text{le}}{{\text{v}}_{ > 0}}\mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \omega \left( { \cdot ;\bar \theta, \lambda, \beta } \right) \supseteq \mathscr{H} $

    因此,式(4) 成立.

    注1  若函数$\bar{\omega }\left( u, v, \tau, \bar{\theta }, \lambda, \beta \right)=\bar{\theta }u-\lambda {{\xi }_{e, D}}\left( \tau \right)\left( \lambda, \beta \in {{\mathbb{R}}_{+}} \right)$,那么等式(3) 也成立.

    定理2.1  函数Pi(x)(i=1,2) 是变分不等式(1) 的间隙函数.

      只需证P2(x)是变分不等式(1) 的间隙函数,P1(x)的证明类似.任取$x\in {{\mathbb{R}}^{n}}$,下面分为两种情形来讨论:

    情形1  设xS,则有$\mathscr{K}\left( x \right)\cap \mathscr{H}=\varnothing $.又由式(3) 知

    $ \mathscr{H} = {\text{le}}{{\text{v}}_{ > 0}}\mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta \left( \cdot \right) + \mathop {\sup }\limits_{z \in \left\{ \cdot \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \cdot \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] $

    于是,

    $ \mathscr{K}\left( x \right) \cap {\text{le}}{{\text{v}}_{ > 0}}\mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }}\\ \left[{\bar \theta \left( \cdot \right) + \mathop {\sup }\limits_{z \in \left\{ \cdot \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \cdot \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] = Ø $

    $ \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] \leqslant 0\;\;\;\;\;\;\\\forall \left( {u, v, \tau } \right) \in \mathscr{K}\left( x \right) $

    因此

    $ \begin{gathered} {P_2}\left( x \right) = \mathop {\sup }\limits_{\left( {{\text{u}}, {\text{v}}, {\tau }} \right) \in \mathscr{K}\left( x \right)} \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + } \right. \hfill \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] \leqslant 0 \hfill \\ \end{gathered} $ (7)

    另一方面,由$ {{0}_{{{\mathbb{R}}_{m}}}}\in g\left( f\left( x \right) \right)-D$

    $ \begin{align} & {{P}_{2}}\left( x \right)=\underset{\left( \text{u},\text{v},\tau \right)\in \mathscr{K}\left( x \right)}{\mathop{\sup }}\,\underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\, \\ & \left[ \bar{\theta }u+\underset{z\in \left\{ v \right\}-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right)-r\sigma \left( z \right) \right)+\underset{t\in \left\{ \tau \right\}-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right)-r\sigma \left( t \right) \right) \right]= \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \underset{y\in {{\mathbb{R}}^{n}}}{\mathop{\sup }}\,\underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }\left( {{\left( f\left( x \right)-y \right)}^{\text{T}}}x \right)+\underset{z\in g\left( y \right)-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right)-r\sigma \left( z \right) \right)+ \right. \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \left. \underset{t\in g\left( f\left( x \right) \right)-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right)-r\sigma \left( t \right) \right) \right]\ge \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }\times 0+\underset{z\in g\left( f\left( x \right) \right)-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right)-r\sigma \left( z \right) \right)+ \right. \\ & \left. \ \ \ \ \ \ \ \ \ \ \ \ \ \underset{t\in g\left( f\left( x \right) \right)-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right)-r\sigma \left( t \right) \right) \right]\ge \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ -\left( \lambda +\beta \right){{\xi }_{e,D}}\left( {{0}_{{{\mathbb{R}}^{m}}}} \right)-2r\sigma \left( {{0}_{{{\mathbb{R}}^{m}}}} \right) \right]=0 \\ \end{align} $ (8)

    从而由(7) 和(8) 知,P2(x)=0.

    反之,假设存在$x\in {{\mathbb{R}}^{n}}$使得P2(x)=0.则有

    $ \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] \leqslant 0\\\;\;\;\;\;\;\;\;\forall \left( {u, v, \tau } \right) \in \mathscr{K}\left( x \right) $

    由(3) 知

    $ \mathscr{H} = {\text{le}}{{\text{v}}_{ > 0}}\mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta \left( \cdot \right) + \mathop {\sup }\limits_{z \in \left\{ \cdot \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \cdot \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] $

    $ \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }} \left[{\bar \theta u + \mathop {\sup }\limits_{z \in \left\{ v \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \tau \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] > 0 $

    对任意$\left( u, v, \tau \right)\in \mathscr{H}$.因此,$\mathscr{K}\left( x \right)\cap \mathscr{H}=\varnothing $成立,即xS.

    情形2  若$x\in {{\mathbb{R}}^{n}}\backslash S$,即xS,那么系统(2) 可行.于是,存在${{y}^{*}}\in {{\mathbb{R}}^{n}}$使得$\left( {{\left( f\left( x \right)-{{y}^{*}} \right)}^{\text{T}}}x, g\left( {{y}^{*}} \right), g\left( f\left( x \right) \right) \right)\in \mathscr{H}$.由(3) 式中的等式知

    $ \bigcap\limits_{\lambda, \beta \in {\mathbb{R}_ + }} {{\text{le}}{{\text{v}}_{ > 0}}\left[{\bar \theta \left( \cdot \right) + \mathop {\sup }\limits_{z \in \left\{ \cdot \right\}-D} \left( {-\lambda {\xi _{e, D}}\left( z \right)-r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in \left\{ \cdot \right\} - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right]} = \mathscr{H} $

    于是对$ \forall \lambda, \beta \in {{\mathbb{R}}_{+}}$,有

    $ \bar \theta \left( {{{\left( {f\left( x \right) - {y^ * }} \right)}^{\text{T}}}x} \right) + \mathop {\sup }\limits_{z \in g\left( {{y^ * }} \right) - D} \left( { - \lambda {\xi _{e, D}}\left( z \right) - r\sigma \left( z \right)} \right) +\\ \mathop {\sup }\limits_{t \in g\left( {f\left( x \right)} \right) - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right) > 0 $

    这意味着

    $ \begin{align} & \underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }\left( {{\left( f\left( x \right)-{{y}^{*}} \right)}^{\text{T}}}x \right)+\underset{z\in g\left( {{y}^{*}} \right)-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right)-r\sigma \left( z \right) \right)+ \right. \\ & \left. \underset{t\in g\left( f\left( x \right) \right)-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right)-r\sigma \left( t \right) \right) \right]\ge 0 \\ \end{align} $

    因此P2(x)≥0.结合情形1和情形2可得${{P}_{2}}\left( x \right)\ge 0\left( \forall x\in {{\mathbb{R}}^{n}} \right) $.

    下面的例子说明,定理2.1是可行的.

    例1  设θ>0,$D=\mathbb{R}_{+}^{3}$.令

    $ \begin{array}{*{20}{c}} {f\left( x \right) = \left\{ \begin{array}{l} - {x^2}\;\;\;\;x < 0\\ \;\;x\;\;\;\;\;\;x \ge 0 \end{array} \right.}\\ {g\left( y \right) = {{\left( {y - {y^2}, y - {y^2}, {y^2} - y} \right)}^{\rm{T}}}} \end{array} $

    则有Ω={0,1}.通过计算得出

    $ {P_1}\left( x \right) = {P_2}\left( x \right) = \left\{ \begin{array}{l} -\bar \theta {x^2}\;\;\;\;x < 0\\ \;\;\bar\theta x\;\;\;\;\;\;x \ge 0 \end{array} \right. $

    因此,由定理2.1知Pi(x)(i=1,2) 是变分不等式(1) 的间隙函数.

    3 误差界

    本节利用上节得到的间隙函数,证明了在逆强伪单调假设条件下,变分不等式(1) 的解集满足误差界.设$ \Omega '=\left\{ x\in {{\mathbb{R}}^{n}}:f\left( x \right)\in \Omega \right\}$.

    定义3.1  设$f:{{\mathbb{R}}^{n}}\to {{\mathbb{R}}^{n}}$为一一映射.称函数f是逆强伪单调的,若存在常数μ>0使得

    $ {\left( {f\left( x \right) - f\left( y \right)} \right)^{\text{T}}}y \geqslant 0 \Rightarrow {\left( {f\left( x \right) - f\left( y \right)} \right)^{\text{T}}}x \geqslant \mu {\left\| {f\left( x \right) - f\left( y \right)} \right\|^2}\;\;\;\;\;\;\;\;\forall x, y \in {\mathbb{R}^n} $

    定义3.2  称函数$f:{{\mathbb{R}}^{n}}\to {{\mathbb{R}}^{n}}$是扩张的,若

    $ \left\| {f\left( x \right) - f\left( y \right)} \right\| \geqslant \left\| {x - y} \right\|\;\;\;\;\;\;\;\;\forall x, y \in {\mathbb{R}^n} $

    命题3.1  设θ>0,则对任意$x\in {{\mathbb{R}}^{n}}, {{P}_{2}}\left( x \right)\le {{P}_{1}}\left( x \right) $.

      由命题1.1可得

    $ \mathop {\sup }\limits_{z \in \left\{ v \right\} - D} \left( { - \lambda {\xi _{e, D}}\left( z \right)} \right) = - \lambda {\xi _{e, D}}\left( v \right) $ (9)

    事实上,任取z∈{v}-D,则有

    $ {\xi _{e, D}}\left( v \right) \leqslant {\xi _{e, D}}\left( z \right) $

    $ - \lambda {\xi _{e, D}}\left( z \right) \leqslant - \lambda {\xi _{e, D}}\left( v \right) $

    于是,等式(9) 成立.同理可得

    $ \mathop {\sup }\limits_{z \in \left\{ v \right\} - D} \left( { - \beta {\xi _{e, D}}\left( z \right)} \right) = - \beta {\xi _{e, D}}\left( v \right) $

    另外,由$ \sigma \left( z \right)\ge 0\left( \forall z\in {{\mathbb{R}}^{n}} \right)$可得,对任意$x\in {{\mathbb{R}}^{n}}$,有

    $ \begin{align} & {{P}_{2}}\left( x \right)=\underset{\left( \text{u},\text{v},\tau \right)\in \mathscr{K}\left( x \right)}{\mathop{\sup }}\,\underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }u+\underset{z\in \left\{ v \right\}-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right)-r\sigma \left( z \right) \right)+ \right. \\ & \left. \underset{t\in \left\{ \tau \right\}-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right)-r\sigma \left( t \right) \right) \right]\le \\ & \ \ \ \ \ \ \ \ \ \ \ \ \underset{\left( \text{u},\text{v},\tau \right)\in \mathscr{K}\left( x \right)}{\mathop{\sup }}\,\underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }u+\underset{z\in \left\{ v \right\}-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right) \right)+\underset{t\in \left\{ \tau \right\}-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right) \right) \right]= \\ & \ \ \ \ \ \ \ \ \ \ \ \ \underset{\left( \text{u},\text{v},\tau \right)\in \mathscr{K}\left( x \right)}{\mathop{\sup }}\,\underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }u-\lambda {{\xi }_{e,D}}\left( v \right)-\beta {{\xi }_{e,D}}\left( \tau \right) \right]={{P}_{1}}\left( x \right) \\ \end{align} $

    定理3.1  设θ>0,S≠Ø,函数f是扩张的且在Ω′上关于μ>0是逆强伪单调的,则对任意x∈Ω′,有

    $ d\left( {x, S} \right) \le \sqrt {\frac{{{P_2}\left( x \right)}}{{\bar \theta \mu }}} $ (10)

      令x*S,则有f(x*)∈Ω.于是对任意y∈Ω′,即f(y)∈Ω,有

    $ {\left( {f\left( y \right) - f\left( {{x^ * }} \right)} \right)^{\rm{T}}}{x^ * } \ge 0 $

    再由fΩ′上关于μ>0的逆强伪单调性知

    $ {\left( {f\left( y \right) - f\left( {{x^ * }} \right)} \right)^{\rm{T}}}y \ge \mu {\left\| {f\left( y \right) - f\left( {{x^ * }} \right)} \right\|^2}\;\;\;\;\;\;\forall y \in \Omega ' $ (11)

    对任意x∈Ω′,有

    $ {P_2}\left( x \right) = \mathop {\sup }\limits_{y \in {\mathbb{R}^n}} \mathop {\inf }\limits_{\lambda, \beta \in {\mathbb{R}_ + }}\\ \left[{\bar \theta \left( {{{\left( {f\left( x \right)-y} \right)}^{\text{T}}}x} \right) + \mathop {\sup }\limits_{z \in g\left( y \right)-D} \left( {-\lambda {\xi _{e, D}}\left( z \right) - r\sigma \left( z \right)} \right) + \mathop {\sup }\limits_{t \in g\left( {f\left( x \right)} \right) - D} \left( { - \beta {\xi _{e, D}}\left( t \right) - r\sigma \left( t \right)} \right)} \right] $

    在上式中令$ y=f\left( {{x}^{*}} \right)\in \Omega \subseteq {{\mathbb{R}}^{n}}$,可得

    $ \begin{align} & {{P}_{2}}\left( x \right)\ge \underset{\lambda ,\beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\,\left[ \bar{\theta }\left( {{\left( f\left( x \right)-f\left( {{x}^{*}} \right) \right)}^{\text{T}}}x \right)+\underset{z\in g\left( f\left( {{x}^{*}} \right) \right)-D}{\mathop{\sup }}\,\left( -\lambda {{\xi }_{e,D}}\left( z \right)-r\sigma \left( z \right) \right)+ \right. \\ & \left. \underset{t\in g\left( f\left( x \right) \right)-D}{\mathop{\sup }}\,\left( -\beta {{\xi }_{e,D}}\left( t \right)-r\sigma \left( t \right) \right) \right]\ge \\ & \ \ \ \ \ \ \ \ \ \ \ \bar{\theta }\left( {{\left( f\left( x \right)-f\left( {{x}^{*}} \right) \right)}^{\rm{T}}}x \right)+\underset{\lambda, \beta \in {{\mathbb{R}}_{+}}}{\mathop{\inf }}\, \left[-\left( \lambda +\beta \right){{\xi }_{e, D}}\left( {{0}_{{{\mathbb{R}}^{m}}}} \right)-2r\sigma \left( {{0}_{{{\mathbb{R}}^{m}}}} \right) \right]= \\ & \ \ \ \ \ \ \ \ \ \ \ \bar{\theta }\left( {{\left( f\left( x \right)-f\left( {{x}^{*}} \right) \right)}^{\rm{T}}}x \right)\ge \\ & \ \ \ \ \ \ \ \ \ \ \ \bar{\theta }\mu {{\left\| f\left( x \right)-f\left( {{x}^{*}} \right) \right\|}^{2}}\ge \\ & \ \ \ \ \ \ \ \ \ \ \ \bar{\theta }\mu {{\left\| x-{{x}^{*}} \right\|}^{2}}\ \ \ \ \ \ \ \ \forall x\in {\Omega }' \\ \end{align} $

    其中第二个不等式的依据是$ {{0}_{{{\mathbb{R}}_{m}}}}\in \left( g\left( f\left( {{x}^{*}} \right) \right)-D \right)\cap \left( g\left( f\left( x \right) \right)-D \right)$,最后一个不等式依据是f扩张.故

    $ d\left( {x, S} \right) \le \left\| {x - {x^ * }} \right\| \le \sqrt {\frac{{{P_2}\left( x \right)}}{{\bar \theta \mu }}} \;\;\;\;\;\;\forall x \in \Omega ' $

    因此,不等式(10) 成立.

    下面的例子说明,定理3.1中f的逆强伪单调性是必要的.

    例3.1  设$\bar{\theta }>0, D=\mathbb{R}_{+}^{3}$.令f(x)=x3g(y)=(yyy+1)T.于是有Ω=Ω′=[0,+∞[.先证fΩ′上关于μ>0的逆强伪单调不成立.事实上,假设(x3-y3)y≥0,那么xy≥0.若fΩ′上关于μ>0是逆强伪单调的,则有

    $ \left( {{x^3}- {y^3}} \right)x \ge \mu {\left( {{x^3}- {y^3}} \right)^2}\;\;\;\;\;\forall x > y \ge 0 $

    这意味着

    $ \frac{x}{{{x^3}- {y^3}}} \ge \mu \;\;\;\;\;\;\forall x > y \ge 0 $

    固定y,于是有μ≤0,x→+∞,这与μ>0矛盾.通过计算可得

    $ {P_2}\left( x \right) = \bar \theta {x^4}\;\;\;\;\forall x \in \Omega '\;\;\;S = \left\{ 0 \right\} $

    下证变分不等式(1) 关于函数P2不满足误差界.给定$m>0, x\left( m \right)=\frac{1}{{{m}^{2}}+1}$.于是有

    $ \sqrt {{P_2}\left( {x\left( m \right)} \right)} = {{\bar \theta }^{\frac{1}{2}}}{\left( {\frac{1}{{{m^2} + 1}}} \right)^2} < {{\bar \theta }^{\frac{1}{2}}}\frac{1}{{{m^2}\left( {{m^2} + 1} \right)}} = \frac{{{{\bar \theta }^{\frac{1}{2}}}}}{{{m^2}}}d\left( {x\left( m \right), S} \right) $

    故(10) 式不成立.

    推论3.1  设θ>0,S≠Ø,函数f是扩张的,并且f在Ω′上关于μ>0是逆强伪单调的,则对任意x∈Ω′,有

    $ d\left( {x, S} \right) \le \sqrt {\frac{{{P_1}\left( x \right)}}{{\bar \theta \mu }}} $

      结合命题3.1和定理3.1立即得出结论.

    下面的例子说明,定理3.1和推论3.1是可行的.

    例3.2  本例沿用例2.1的假设.通过计算得Ω′={0,1}.显然,函数f在Ω′上关于μ=1是逆强伪单调的.由例2.1知

    $ {P_i}\left( x \right) = \bar \theta {x^2}\;\;\;\;\;i = 1, 2\;\;\;\;\;\forall x \in \Omega '\;\;\;\;\;S = \left\{ 0 \right\} $

    故有

    $ d\left( {x, S} \right) = \left\| x \right\| \le \sqrt {\frac{{{P_i}\left( x \right)}}{{\bar \theta \mu }}} \;\;\;\;\;\;i = 1, 2\;\;\;\;\;\forall x \in \Omega ' $
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    Gap Functions and Error Bounds for a Class of Variational Inequalities with Cone Constraints
    DONG Wen1, OU Xiao-qing2, LI Jing-fu1, CHEN Jia-wei1     
    1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
    2. School of Management, Chongqing College of Humanities, Science & Technology, Chongqing 401524, China
    Abstract: The gap function and the error bound play an important role in optimization methods and the error bound, especially, can characterize the effective estimated distance between a feasible point and the solution set of variational inequalities. In this article, by using the image space analysis, gap functions for a class of variational inequalities with cone constraints are proposed. Moreover, error bounds, which provide an effective estimated distance between a feasible point and the solution set, for the variational inequalities are established via the gap functions.
    Key words: constrained variational inequality    image space analysis    gap function    error bound    
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