西南师范大学学报(自然科学版)   2019, Vol. 44 Issue (7): 17-22.  DOI: 10.13718/j.cnki.xsxb.2019.07.003
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  • Banach空间中集值隐函数的类Lipschitz性质及其应用    [PDF全文]
    肖成英 1, 杨明歌 2     
    1. 四川工商学院 云计算与智能信息处理重点实验室, 成都 611745;
    2. 上海大学 管理学院, 上海 200444
    摘要:利用变分分析和广义微分的相关工具,在Banach空间中研究集值隐函数的稳定性,给出集值隐函数在给定点具有类Lipschitz性质的Clarke上导数充分条件.作为应用,讨论参数向量优化问题有效解映射的稳定性,给出有效解映射在给定点具有类Lipschitz性质的Clarke上导数充分条件.所得结果改进了相关文献中的结果.
    关键词集值隐函数    稳定性分析    参数向量优化问题    有效解映射    类Lipschitz性质    

    集值隐函数的稳定性已被广泛研究[1-7].最近,文献[1]研究集值隐函数的类Lipschitz性质,利用Fréchet上导数和Mordukhovich上导数,在Asplund空间中给出集值隐函数在给定点具有类Lipschitz性质的充分条件,由此导出参数向量优化问题的有效解映射在给定点具有类Lipschitz性质的充分条件.虽然文献[1]定理3.1在适当的条件下证明了集值隐函数在给定点是类Lipschitz的,但是该定理的证明过程并不严谨.作者采用的是反证法,为了导出矛盾,需要证明:存在(x*y*)和(x3y3)满足0∉Fp(x3),但不满足条件(ⅲ).遗憾的是,0∉Fp(x3)的证明被遗漏了,且难以从文献[1]定理3.1的假设条件推出.为了解决这个问题,本文采用不同于文献[1]定理3.1的证明方法,给出集值隐函数在给定点是类Lipschitz的严格证明,证明过程避开了0∉Fp(x3).

    XY是Banach空间,P是度量空间,FP×XY是集值映射,定义集值隐函数GPX如下:

    $ G(p) :=\{x \in X | 0 \in F(p, x)\} $ (1)

    下面讨论(1)式中的集值隐函数G的类Lipschitz性质,给出G在给定点类Lipschitz性质成立的上导数充分条件.使用记号Fp(·):=F(p,·).

    定理 1    设XY是Banach空间,P是度量空间,FP×XY是集值映射,GPX是由(1)式定义的集值隐函数,(px)∈P×X且0∈F(px).若存在常数r>0满足下列条件:

    (ⅰ)任意的pB(pr),集值映射Fp是闭的;

    (ⅱ)常数

    $ \begin{array}{l} {k_r}: = \mathop {\lim }\limits_{\delta \downarrow 0} \inf \left\{ {\left\| {{x^*}} \right\||{x^*} \in D_c^*{F_p}(x,y)\left( {{y^*}} \right),p \in B(\bar p,r),x \in B(\bar x,r)\backslash G(p)} \right.,\\ \;\;\;\;\;\;\;\;y \in \prod\nolimits_\delta {\left( {0;{F_p}(x)} \right)} \cap B(0,r),{y^*} \in {J_\delta }(y)\} > 0 \end{array} $

    其中:任意δ>0,∏δ(0;Fp(x)):={yFp(x)| ||y||<d(0,Fp(x))+δ},Jδ(y):={y*SY*|d(y*J(y))<δ};

    (ⅲ)存在常数l>0使得

    $ F\left( {p',x} \right) \cap r{B_Y} \subset F(p,x) + ld\left( {p',p} \right){B_Y},\forall x \in B(\bar x,r),\forall p,p' \in B(\bar p,r) $

    G在(px)是类Lipschitz的且具有系数$\frac{l}{k_{r}}$.

        任给常数r>0和l>0,假设满足定理1的条件.首先证明:任意的μ∈(0,min{rkrr}),若$(x, p) \in B\left(\overline{x}, r-\frac{\mu}{k_{r}}\right) \times B(\overline{p}, r)$d(0,F(px))<μ,则

    $ d\left( {x,G\left( p \right)} \right) \le \frac{1}{{{k_r}}}d\left( {0,F\left( {p,x} \right)} \right) $ (2)

    采用反证法,假设(2)式不成立,则存在μ∈(0,min{rkrr}),$x_{0} \in B\left(\overline{x}, r-\frac{\mu}{k_{r}}\right)$$p_{0} \in B(\overline{p}, r)$,满足$d\left(0, F\left(p_{0}, x_{0}\right)\right)<\mu$,但

    $ d\left( {{x_0},G\left( {{p_0}} \right)} \right) > \frac{{d\left( {0,F\left( {{p_0},{x_0}} \right)} \right)}}{{{k_r}}} $

    从而x0G(p0),即0∉F(p0x0).由条件(ⅰ),d(0,F(p0x0))>0.令ε:=d(0,F(p0x0)),则ε∈(0,μ).进一步,根据实数的稠密性,选取$k \in\left(\frac{\varepsilon k_{r}}{\mu}, k_{r}\right)$满足

    $ d\left( {{x_0},G\left( {{p_0}} \right)} \right) > \frac{{d\left( {0,F\left( {{p_0},{x_0}} \right)} \right)}}{k} > \frac{{d\left( {0,F\left( {{p_0},{x_0}} \right)} \right)}}{{{k_r}}} $ (3)

    λ:=εk-1,则由(3)式得

    $ d\left(x_{0}, G\left(p_{0}\right)\right)>\frac{d\left(0, F\left(p_{0}, x_{0}\right)\right)}{k}=\lambda $ (4)

    由距离函数的定义,任意的α∈(0,r-μ),存在y0F(p0x0),使得

    $ \left\|y_{0}\right\|<d\left(0, F\left(p_{0}, x_{0}\right)\right)+\alpha=\varepsilon+\alpha<\mu+\alpha<r $ (5)

    定义函数φ$X \times Y \longrightarrow \overline{\mathbb{R}}$如下:

    $ \varphi \left( {x,y} \right): = \left\| y \right\| + {\delta _{{\rm{gph}}}}{F_{{p_0}}}\left( {x,y} \right),\forall \left( {x,y} \right) \in X \in Y $

    由条件(ⅰ),φX×Y上是下半连续的.由(5)式得

    $ \varphi \left( {{x_0},{y_0}} \right) = \left\| {{y_0}} \right\| < \varepsilon + \alpha + \mathop {\inf }\limits_{\left( {x,y} \right) \in X \times Y} \varphi \left( {x,y} \right) $

    任意的$\eta \in\left(0, \frac{\lambda}{\varepsilon+\alpha}\right)$,在乘积空间X×Y中使用范数$\|(x, y)\|_{\eta} :=\|x\|+\eta\|y\|$.由文献[9]定理2.26中的Ekeland变分原理,存在$(\mathit{\hat x}, \mathit{\hat y})$X×Y满足

    $ \varphi \left( {\hat x,\hat y} \right) \le \varphi \left( {{x_0},{y_0}} \right),{\left\| {\left( {\hat x,\hat y} \right) - \left( {{x_0},{y_0}} \right)} \right\|_\eta } \le \lambda $

    $ \varphi \left( {\hat x,\hat y} \right) \le \varphi \left( {{x_0},{y_0}} \right) + \frac{{\varepsilon + \alpha }}{\lambda }{\left\| {\left( {\hat x,\hat y} \right) - \left( {x,y} \right)} \right\|_\eta },\forall \left( {x,y} \right) \in X \times Y $

    从而

    $ \left( {\hat x,\hat y} \right) \in {\rm{gph}}{F_{{p_0}}},\left\| {\hat y} \right\| \le \left\| {{y_0}} \right\| < r,\left\| {\hat x - {x_0}} \right\| + h\left\| {\hat y - {y_0}} \right\| \le \lambda $ (6)

    $ \left\| {\hat y} \right\| \le \left\| y \right\| + \frac{{\varepsilon + \alpha }}{\lambda }\left( {\left\| {x - \hat x} \right\| + \eta \left\| {y - \hat y} \right\|} \right) + {\delta _{{\rm{gph}}{F_{{p_0}}}}}\left( {x,y} \right),\forall \left( {x,y} \right) \in X \times Y $ (7)

    由(4)式和(6)式,$\left\| {\hat x - {x_0}} \right\| \le \lambda < d\left( {{x_0}, G\left( {{p_0}} \right)} \right)$,从而$\stackrel{\wedge}{x} \notin G\left(p_{0}\right)$,即0$\notin F\left(p_{0}, \hat{x}\right)$,故$\hat{y} \neq 0$.进一步,由(6)式和k的取法可知

    $ \left\| {\hat x - \bar x} \right\| \le \left\| {\hat x - {x_0}} \right\| + \left\| {{x_0} - \bar x} \right\| \le \lambda + r - \mu {k_r}^{ - 1} = \varepsilon {k^{ - 1}} + r - \mu {k_r}^{ - 1} < r $

    定义函数ψ$X \times Y \longrightarrow \overline{\mathbb{R}}$如下:

    $ \psi \left( {x,y} \right): = \left\| y \right\| + \frac{{\varepsilon + \alpha }}{\lambda }\left( {\left\| {x - \hat x} \right\| + \eta \left\| {y - \hat y} \right\|} \right),\forall \left( {x,y} \right) \in X \times Y $

    由(7)式易知,$(\mathit{\hat x}, \mathit{\hat y})$是函数$\psi + {\delta _{{\rm{gph}}{\mathit{F}_{{p_{_0}}}}}}$X×Y上的极小值点.注意到$\hat{y} \neq 0$,由文献[8]命题1.114得

    $ \begin{array}{l} \left( {0,0} \right) \in {\partial _c}\psi \left( {\hat x,\hat y} \right) + {\partial _c}{\delta _{{\rm{gph}}}}{F_{{p_0}}}\left( {\hat x,\hat y} \right) = \\ \;\;\;\;\;\;\;\;\;\;\left\{ 0 \right\} \times J\left( {\hat y} \right) + \frac{{\varepsilon + \alpha }}{\lambda }\left( {{B_{{X^ * }}} \times \eta {B_{{Y^ * }}}} \right) + {N_c}\left( {\left( {\hat x,\hat y} \right);{\rm{gph}}{F_{{p_0}}}} \right) \end{array} $

    则存在$y_1^* \in J(\hat y)$和(x2*y2*)∈BX*×BY*使得

    $ \left( { - \frac{{\varepsilon + \alpha }}{\lambda }x_2^ * , - y_1^ * - \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right) \in {N_c}\left( {\left( {\hat x,\hat y} \right);{\rm{gph}}{F_{{p_0}}}} \right) $

    显然,

    $ \left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\| \ge 1 - \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }\left\| {y_2^ * } \right\| \ge 1 - \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda } > 0 $

    $ {\bar x^ * }: = \frac{{ - \frac{{\varepsilon + \alpha }}{\lambda }x_2^ * }}{{\left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|}},{\tilde y^ * }: = \frac{{y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * }}{{\left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|}} $

    $\left( {{{\bar x}^*}, - {{\bar y}^*}} \right) \in {N_c}\left( {(\mathit{\hat x}, \hat y);{\mathop{\rm gph}\nolimits} {F_{{p_0}}}} \right)$,故${\tilde x^*} \in D_c^*{F_{{p_0}}}(\mathit{\hat x}, \mathit{\hat y})\left( {{{\tilde y}^*}} \right)$.任意的$y \in {F_{{p_{_0}}}}(\mathit{\hat x})$,由(7)式得

    $ \left\| {\hat y} \right\| \le \left\| y \right\| + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }\left\| {y - \hat y} \right\| \le \left[ {1 + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }} \right]\left\| y \right\| + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }\left\| {\hat y} \right\| $

    从而

    $ \left\| {\hat y} \right\| \le \left[ {1 + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }} \right]d\left( {0,{F_{{p_0}}}\left( {\hat x} \right)} \right) + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }\left\| {\hat y} \right\| $

    $ \left\| {\hat y} \right\| < d\left( {0,{F_{{p_0}}}\left( {\hat x} \right)} \right) + \frac{{2\left( {\varepsilon + \alpha } \right)\eta }}{{\lambda - \left( {\varepsilon + \alpha } \right)\eta }}d\left( {0,{F_{{p_0}}}\left( {\hat x} \right)} \right) $ (8)

    进一步,

    $ \left\| {{{\tilde x}^ * }} \right\| = \frac{{\left\| {\frac{{\varepsilon + \alpha }}{\lambda }x_2^ * } \right\|}}{{\left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|}} \le \frac{{\left( {\varepsilon + \alpha } \right)k}}{\varepsilon }{\left[ {1 - \frac{{\left( {\varepsilon + \alpha } \right)k\eta }}{\varepsilon }} \right]^{ - 1}} $ (9)

    由于

    $ 1 - \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda } \le \left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\| \le 1 + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda } $

    $ \left| {1 - \left\| {{y^ * } + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|} \right| \le \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda } $

    从而

    $ \begin{array}{l} \left\| {{{\tilde y}^ * } - y_1^ * } \right\| = \left\| {\frac{{y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * }}{{\left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|}} - y_1^ * } \right\| = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\left\| {\left[ {1 - \left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|} \right]y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|}}{{\left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\left| {1 - \left\| {y_1^ * + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }y_2^ * } \right\|} \right| + \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }}}{{1 - \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }}} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\frac{{2\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }}}{{1 - \frac{{\left( {\varepsilon + \alpha } \right)\eta }}{\lambda }}} = \frac{{2\left( {\varepsilon + \alpha } \right)\eta }}{{\lambda - \left( {\varepsilon + \alpha } \right)\eta }} \end{array} $ (10)

    对任意的δ>0,只要上述的αη选取得充分小,由(8),(9)和(10)式可得

    $ \hat y \in \prod\nolimits_\delta {\left( {0;{F_{{p_0}}}\left( {\hat x} \right)} \right)} ,\left\| {{{\tilde x}^ * }} \right\| < k + \delta ,{\tilde y^ * } \in {J_\delta }\left( {\hat y} \right) $

    δ↓0,则$\left\|\tilde{x}^{*}\right\| \leqslant k<k_{r}$,这与条件(ⅱ)矛盾,故(2)式成立.

    其次,任取$\rho \in\left(0, r-\frac{\mu}{k_{r}}\right)$且2μ,下面证明

    $ G\left( {p'} \right) \cap B\left( {\bar x,\rho } \right) \subset G\left( p \right) + \frac{l}{{{k_r}}}d\left( {p',p} \right){B_X},\forall p',p \in B\left( {\bar p,\rho } \right) $ (11)

    事实上,任意的p′,pB(pρ),任意的xG(p′)∩B(xρ),显然0∈F(p′,x).由条件(ⅲ),0∈F(px)+ld(p′,p)BY,故

    $ d\left( {0,F\left( {p,x} \right)} \right) \le ld\left( {p',p} \right) \le l\left[ {d\left( {p',\bar p} \right) + d\left( {\bar p,p} \right)} \right] \le 2l\rho < \mu $ (12)

    由(2)和(12)式得

    $ d\left( {x,G\left( p \right)} \right) \le \frac{1}{{{k_r}}}d\left( {0,F\left( {p,x} \right)} \right) \le \frac{1}{{{k_r}}}d\left( {p',p} \right) $

    $ x \in G\left( p \right) + \frac{l}{{{k_r}}}d\left( {p',p} \right){B_X} $

    即(11)式成立.因此,G在(px)是类Lipschitz的且具有系数$\frac{l}{k_{r}}$.

    注 1     定理1利用Clarke上导数,在Banach空间中给出集值隐函数在给定点具有类Lipschitz性质的充分条件,而文献[1]定理3.1利用Fréchet上导数,在Asplund空间中给出集值隐函数在给定点具有类Lipschitz性质的充分条件.定理1中的上导数条件(ⅱ)与文献[1]定理3.1中的上导数条件(ⅲ)不同,定理1的证明方法与文献[1]定理3.1的证明方法也不同.由于采用了不同的证明方法,避免了证明0∉Fp(x3),而在文献[1]定理3.1的证明中,0∉Fp(x3)是必要的,但却被遗漏了.进一步,由定理1可知,若文献[6]定理7中的Jδ(y)定义为Jδ(y):={y*SY*|d(y*J(y))<δ},则结论仍然成立.

    下面讨论集值隐函数的类Lipschitz性质在参数向量优化问题有效解映射的稳定性分析中的应用.设XY是Banach空间,P是度量空间,fP×X→Y是向量值函数,KY是顶点在原点的尖闭凸锥,AYyA,则称yA关于K的有效点(efficient point)当且仅当(y-K)∩A={y}.A的有效点的集合记为EffKA,规定EffKØ=Ø.

    考虑参数向量优化问题:

    $ {\rm{Ef}}{{\rm{f}}_K}\left\{ {f\left( {p,x} \right)\left| {x \in X} \right.} \right\} $ (13)

    其中:x是未知的决策变量,p是参数.任意的pP,令

    $ \mathscr{F}\left( p \right):{\rm{Ef}}{{\rm{f}}_K}\left\{ {f\left( {p,x} \right)\left| {x \in X} \right.} \right\} $

    $ \mathscr{S}\left( p \right): = \left\{ {x \in X\left| {f\left( {p,x} \right) \in \mathscr{F}\left( p \right)} \right.} \right\} $ (14)

    则称$\mathscr{F} : P \rightrightarrows Y$$\mathscr{S} : P \rightrightarrows X$分别为参数向量优化问题(13)的有效点映射(efficient point multifunction)和有效解映射(efficient solution map).

    由定理1可知,若F满足一定的条件,则集值隐函数G在给定点是类Lipschitz的.在此基础上,可以寻找有效解映射$\mathscr{S}$在给定点具有类Lipschitz性质的充分条件.注意到,有效解映射$\mathscr{S}$可以用集值隐函数来表达,即

    $ \mathscr{S}(p) :=\{x \in X | 0 \in H(p, x)\} $

    其中HP×XY定义为

    $ H(p, x) :=-f(p, x)+\mathscr{F}(p) $ (15)

    事实上,H是由目标函数f和有效点映射$\mathscr{F}$构成的集值映射.为清楚起见,记Hp(·):=H(p,·).

    定理 2     设XY是Banach空间,P是度量空间,HP×XY是由(15)式定义的集值映射,$\mathscr{S}$PX是由(14)式定义的有效解映射,(px)∈P×X且(px)∈gph$\mathscr{S}$.若存在常数r>0满足下列条件:

    (ⅰ)任意的pB(pr),集值映射Hp是闭的;

    (ⅱ)常数

    $ \begin{array}{*{20}{c}} {{k_r}: = \mathop {\lim }\limits_{\delta \downarrow 0} \inf \left\{ {\left\| {{x^ * }} \right\|\left| {{x^ * } \in D_c^ * {H_p}\left( {x,y} \right)\left( {{y^ * }} \right),p \in B\left( {\bar p,r} \right),x \in B\left( {\bar x,r} \right)\backslash \mathscr{S}\left( p \right),} \right.} \right.}\\ {\left. {y \in \prod\nolimits_\delta {\left( {0;{H_p}\left( x \right)} \right)} \cap B\left( {0,r} \right),{y^ * } \in {J_\delta }\left( y \right)} \right\} > 0} \end{array} $

    其中:任意δ>0, $\prod\limits_{} {{}_\delta \left( {0;{H_p}(x)} \right)} : = \left\{ {y \in {H_\rho }(x)|\left\| y \right\| < d\left( {0, {H_p}(x)} \right) + \delta } \right\}, {J_\delta }(y): = \left\{ {{y^*} \in {S_{{Y^ * }}}} \right.\left| {d\left( {{y^*}, J(y)} \right) < \delta \} } \right.$

    (ⅲ)存在常数l>0使得

    $ H\left( {p',x} \right) \cap r{B_Y} \subset H\left( {p,x} \right) + ld\left( {p',p} \right){B_Y},\forall x \in B\left( {\bar x,r} \right),\forall p,p' \in B\left( {\bar p,r} \right) $

    则有效解映射$\mathscr{S}$在(px)是类Lipschitz的且具有系数$\frac{l}{k_{r}}$.

        注意到$(\overline{p}, \overline{x}) \in \operatorname{gph} \mathscr{S} \Leftrightarrow 0 \in H(\overline{p}, \overline{x})$,分别用H$\mathscr{S}$代替定理1中的FG,可得定理2的结论.

    由定理2可知,当H满足一定的条件时,有效解映射$\mathscr{S}$在给定点是类Lipschitz的.而H是由f$\mathscr{F}$构成的,故可以将定理2中的条件(ⅲ)换成由f$\mathscr{F}$刻画的条件,见下面的定理3.

    定理 3     设XY是Banach空间,P是度量空间,HP×XY是由(15)式定义的集值映射,$\mathscr{S}$PX是由(14)式定义的有效解映射,(px)∈P×X且(px)∈gph$\mathscr{S}$.若f在(px)是严格可微的,且存在常数r>0满足下列条件:

    (ⅰ)任意的pB(pr),集值映射Hp是闭的;

    (ⅱ)常数

    $ \begin{array}{*{20}{c}} {{k_r}: = \mathop {\lim }\limits_{\delta \downarrow 0} \inf \left\{ {\left\| {{x^ * }} \right\|\left| {{x^ * } \in D_c^ * {H_p}\left( {x,y} \right)\left( {{y^ * }} \right),p \in B\left( {\bar p,r} \right),x \in B\left( {\bar x,r} \right)\backslash \mathscr{S}\left( p \right)} \right.,} \right.}\\ {\left. {y \in \prod\nolimits_\delta {\left( {0;{H_p}\left( x \right)} \right) \cap B\left( {0,r} \right)} ,{y^ * } \in {J_\delta }\left( y \right)} \right\} > 0} \end{array} $

    其中:任意δ>0, $\prod\limits_{} {{}_\delta \left( {0;{H_p}(x)} \right)} : = \left\{ {y \in {H_\rho }(x)|\left\| y \right\| < d\left( {0, {H_p}(x)} \right) + \delta } \right\}, {J_\delta }(y): = \left\{ {{y^*} \in {S_{{Y^ * }}}} \right.\left| {d\left( {{y^*}, J(y)} \right) < \delta \} } \right.$

    (ⅲ)有效点映射$\mathscr{F}$p是局部Lipschitz的.

    则有效解映射$\mathscr{S}$在(px)是类Lipschitz的.

        因为f在(px)严格可微,所以f在(px)是Lipschitz连续的,故存在x的邻域Up的邻域W1和实数l1>0使得

    $ \left\| {f\left( {p',x} \right) - f\left( {p,x} \right)} \right\| \le {l_1}d\left( {p',p} \right),\forall x \in U,\forall p',p \in {W_1} $ (16)

    由条件(ⅲ),存在p的邻域W2和实数l2>0使得

    $ \mathscr{F}\left( {p'} \right) \subset \mathscr{F}\left( p \right) + {l_2}d\left( {p',p} \right){B_Y},\forall p',p \in {W_2} $ (17)

    W:=W1W2,则

    $ H\left( {p',x} \right) \subset H\left( {p,x} \right) + \left( {{l_1} + {l_2}} \right)d\left( {p',p} \right){B_Y},\forall x \in U,\forall p',p \in W $ (18)

    事实上对任意的xU,任意的p′,pW,任意的yH(p′,x),显然y∈-f(p′,x)+$\mathscr{F}$(p′).结合(17)式得

    $ y \in - f\left( {p',x} \right) + \mathscr{F}\left( p \right) + {l_2}d\left( {p',p} \right){B_Y} $ (19)

    由(16)式得

    $ f\left( {p',x} \right) \in f\left( {p,x} \right) + {l_1}d\left( {p',p} \right){B_Y} $

    $ - f\left( {p',x} \right) \in - f\left( {p,x} \right) + {l_1}d\left( {p',p} \right){B_Y} $ (20)

    由(19)和(20)式可知

    $ y \in - f\left( {p,x} \right) + \mathscr{F}\left( p \right) + \left( {{l_1} + {l_2}} \right)d\left( {p',p} \right){B_Y} $

    $ y \in H(p, x)+\left(l_{1}+l_{2}\right) d\left(p^{\prime}, p\right) B_{Y} $

    注意到yH(p′,x)是任意的,故(18)式成立.综上,定理2的所有条件满足.由定理2可知定理3成立.

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    Lipschitz-Like Property of Implicit Multifunctions in Banach Spaces and Applications
    XIAO Cheng-ying 1, YANG Ming-ge 2     
    1. Key Laboratory of Cloud Computing and Intelligent Information Processing, Sichuan Technology and Business University, Chengdu 611745, China;
    2. School of Management, Shanghai University, Shanghai 200444, China
    Abstract: In this paper, by means of variational analysis and generalized differentiation, we study the stability of implicit multifunctions in Banach spaces, and give sufficient conditions in terms of Clarke coderivative to show that the implicit multifunction is Lipschitz-like at the given point. As applications, we discuss the stability of parametric vector optimization problems, and provide sufficient conditions in terms of Clarke coderivative for guaranteeing the efficient solution map to be Lipschitz-like at the given point. The results obtained in this paper improve the corresponding results in the literature.
    Key words: implicit multifunction    stability analysis    parametric vector optimization problem    efficient solution map    Lipschitz-like property    
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