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Calmness of Solution Mapping for a Class of Parametric Variational Systems
DOI: 10.12677/PM.2023.1310287, PDF, HTML, XML, 下载: 266  浏览: 564

Abstract: This paper discusses the stability properties of parametric variational systems in a finite dimen-sional Euclidean space. Unlike the method of using Fréchet derivatives and graph derivatives to discuss the sufficient conditions for calmness, this paper uses derivative criteria for metric regu-larity and modern variational analysis techniques to provide new sufficient conditions for such parametric variational systems to have calmness at a given point.

1. 引言

$0\in f\left(x\right)+Q\left( x \right)$

(其中f，Q为Banach空间之间的映射，且f为单值的，Q为集值的)及其在扰动参数发生扰动时的局部灵敏性。当 $Q\left(x\right)$ 为凸集C生成的法锥时，则广义方程问题归结为经典的变分不等式问题，即

$x\in C$ ，使得 $〈f\left(x\right),u-x〉\ge 0,\text{\hspace{0.17em}}\forall u\in C$

$0\in f\left(p,x\right)+Q\left(p,x\right),$

(其中p为扰动参数)可以描述驻点和Karush-Kuhn-Tucher (KKT)点的扰动集合。因此，基于广义方程提供的参数扰动下最优解灵敏性分析的模型，适定性的研究对集值分析、优化理论及其应用发展便显得尤为关键，其中静态性质的研究具有非常重要的作用。静态性质的概念，最早由Clarke [1] 为研究特定的优化问题而引入的，发展到今天，静态性质在广义方程的灵敏性分析、变分不等式、必要的最优条件等方面起到了关键性的作用，大量的学者对静态性质都做了许多相关的研究 [2] - [10] 。本文中，主要研究参数变分系统解映射

$S\left(p\right):=\left\{x\in {ℝ}^{n}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}0\in F\left(p,x\right)\right\}$ (1.1)

2. 预备知识

${\delta }_{A}\left(x\right)=\left\{\begin{array}{l}0\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}x\in A.\hfill \\ +\infty \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\notin A.\hfill \end{array}$

$H\left(x\right)\cap V\subseteq H\left(\stackrel{¯}{x}\right)+\kappa ‖x-\stackrel{¯}{x}‖\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in U,$

${T}_{A}\left(x\right):=\left\{w:\exists {t}_{k}↓0,\left\{{w}_{k}\right\}\subset {ℝ}^{n}\text{,}\text{\hspace{0.17em}}满足\text{ }\text{ }{w}_{k}\to w,\text{\hspace{0.17em}}且\text{ }\text{ }x+{t}_{k}{w}_{k}\in A\text{}\forall k\right\}.$

${T}_{A}\left(x\right)$ 的极锥 ${\stackrel{^}{N}}_{A}\left(x\right)$ 为正则法锥。我们用 ${N}_{A}\left(x\right)$ 表示A在x处的极限法锥，定义为

${N}_{A}\left(x\right):=\left\{v:\exists {x}_{k}\to x,\text{\hspace{0.17em}}{v}_{k}\to v,\text{\hspace{0.17em}}满足\text{ }\text{ }{x}_{k}\in A,\text{\hspace{0.17em}}且\text{ }\text{ }{v}_{k}\in {\stackrel{^}{N}}_{A}\left(x\right)\text{}\forall k\right\}$ .

${‖H‖}^{+}:=\underset{x\in \mathbb{B}}{\mathrm{sup}}\underset{u\in H\left(x\right)}{\mathrm{sup}}‖u‖=\mathrm{inf}\left\{\kappa >0\text{\hspace{0.17em}}|\text{\hspace{0.17em}}y\in H\left(x\right)⇒‖y‖\le \kappa ‖x‖\right\}.$

${‖H‖}^{-}:=\underset{x\in \mathbb{B}}{\mathrm{sup}}\underset{u\in H\left(x\right)}{\mathrm{inf}}‖u‖=\mathrm{inf}\left\{\kappa >0\text{\hspace{0.17em}}|\text{\hspace{0.17em}}H\left(x\right)\cap \kappa \mathbb{B}\ne \varnothing \text{\hspace{0.17em}}\forall x\in \mathbb{B}\right\}.$

$z\in DH\left(\stackrel{¯}{x}|\stackrel{¯}{u}\right)\left(v\right)⇔\left(v,z\right)\in {T}_{gph\text{ }H}\left(\stackrel{¯}{x},\stackrel{¯}{u}\right).$

${x}^{*}\in {\stackrel{^}{D}}^{*}H\left(\stackrel{¯}{x}|\stackrel{¯}{u}\right)\left({u}^{*}\right)⇔\left({x}^{*},-{u}^{*}\right)\in {\stackrel{^}{N}}_{gph\text{ }H}\left(\stackrel{¯}{x},\stackrel{¯}{u}\right),$

${x}^{*}\in {D}^{*}H\left(\stackrel{¯}{x}|\stackrel{¯}{u}\right)\left({u}^{*}\right)⇔\left({x}^{*},-{u}^{*}\right)\in {N}_{gph\text{ }H}\left(\stackrel{¯}{x},\stackrel{¯}{u}\right).$

$d\left(\stackrel{¯}{x}\right)\le \underset{x\in E}{\mathrm{inf}}f\left(x\right)+\epsilon .$

(i) $d\left(x,\stackrel{¯}{x}\right)\le \lambda$

(ii) $f\left(x\right)\le f\left(\stackrel{¯}{x}\right)$

(iii) $f\left(x\right)

$d\left(x,{H}^{-1}\left(u\right)\right)\le \kappa d\left(u,H\left(x\right)\right),\text{\hspace{0.17em}}\forall x\in U,u\in V,$

$\underset{\begin{array}{c}\left(x,y\right)\to \left(\stackrel{¯}{x},\stackrel{¯}{y}\right)\\ \left(x,y\right)\in gph\text{ }G\end{array}}{\mathrm{lim}\mathrm{sup}}{‖DG{\left(x|y\right)}^{-1}‖}^{-}={‖{D}^{*}G{\left(\stackrel{¯}{x}|\stackrel{¯}{y}\right)}^{-1}‖}^{+}.$

$\underset{\begin{array}{c}x\to \stackrel{¯}{x}\\ u\to \stackrel{¯}{x}\end{array}}{\mathrm{lim}}\frac{f\left(x\right)-f\left(u\right)-\nabla f\left(\stackrel{¯}{x}\right)\left(x-u\right)}{‖x-u‖}=0.$

3. 参数变分系统解映射的静态性质

(i) $gph\text{\hspace{0.17em}}{F}_{\stackrel{¯}{p}}$ 在点 $\left(\stackrel{¯}{x},0\right)$ 附近是局部闭的；

(ii) 存在 $\mathcal{l}>0$ 以及 $\delta >0$ 使得

$F\left(p,x\right)\subseteq F\left(\stackrel{¯}{p},x\right)+\mathcal{l}d\left(p,\stackrel{¯}{p}\right)\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in {\mathbb{B}}_{\delta }\left(\stackrel{¯}{x}\right),\forall p\in {\mathbb{B}}_{\delta }\left(\stackrel{¯}{p}\right);$

(iii) 对任意正数c，有 $\underset{\begin{array}{c}\left(x,y\right)\to \left(x,0\right)\\ \left(x,y\right)\in gph\text{ }F\end{array}}{\mathrm{lim}\mathrm{sup}}{‖D{F}_{\stackrel{¯}{p}}{\left(x|y\right)}^{-1}‖}^{-}

(i) $gph\text{\hspace{0.17em}}G$ 在点 $\left(\stackrel{¯}{x},\stackrel{¯}{y}\right)$ 附近是局部闭的，

(ii) ${‖{D}^{*}G{\left(\stackrel{¯}{x}|\stackrel{¯}{y}\right)}^{-1}‖}^{+}

$‖{x}_{\nu }-x‖\le \stackrel{˜}{c}‖{y}_{\nu }-y‖.$

$\left(0,v\right)\in {\stackrel{^}{N}}_{gph\text{\hspace{0.17em}}G}\left(x,y\right)⇒v=0.$ (3.1)

${‖{D}^{*}G{\left(\stackrel{¯}{x}|\stackrel{¯}{y}\right)}^{-1}‖}^{+}=\mathrm{inf}\left\{\kappa >0|{y}^{*}\in {D}^{*}G{\left(\stackrel{¯}{x}|\stackrel{¯}{y}\right)}^{-1}\left({x}^{*}\right)⇒‖{y}^{*}‖\le \kappa ‖{x}^{*}‖\right\}$

$\left({x}^{*},-{y}^{*}\right)\in {\stackrel{^}{N}}_{gph\text{\hspace{0.17em}}G}\left(x,y\right)⇒‖{y}^{*}‖\le c‖{x}^{*}‖.$ (3.2)

$〈{y}^{*},\frac{{y}^{\prime }-y}{‖{y}^{\prime }-y‖}〉\le ‖{y}^{*}‖‖\frac{{y}^{\prime }-y}{‖{y}^{\prime }-y‖}‖\le c‖{x}^{*}‖.$ (3.3)

$\left({x}_{\nu },{y}_{\nu }\right)\in gph\text{\hspace{0.17em}}G\cap \left({\mathbb{B}}_{\eta }\left(\stackrel{¯}{x}\right)×{\mathbb{B}}_{\eta }\left(\stackrel{¯}{y}\right)\right)$$\left({x}_{\nu },{y}_{\nu }\right)\ne \left(x,y\right)$ ，使得

$‖{x}_{\nu }-x‖>\stackrel{˜}{c}‖{y}_{\nu }-y‖:=\lambda .$ (3.4)

$\phi \left(u,v\right):=‖v-y‖+{\delta }_{gph\text{ }G\cap \left({\mathbb{B}}_{\eta }\left(\stackrel{¯}{x}\right)×{\mathbb{B}}_{\eta }\left(\stackrel{¯}{y}\right)\right)}\left(u,v\right).$

$gph\text{\hspace{0.17em}}G$ 在点 $\left(\stackrel{¯}{x},\stackrel{¯}{y}\right)$ 附近是局部闭性，容易验证 $\phi$ 是下半连续的， $\mathrm{inf}\phi$ 为有限数，且

$\phi \left({x}_{\nu },{y}_{\nu }\right)\le \mathrm{inf}\phi +\frac{\lambda }{\stackrel{˜}{c}}.$

$h\left(\stackrel{˜}{x}-{x}_{\nu },\stackrel{˜}{y}-{y}_{\nu }\right)\le \frac{c+\stackrel{˜}{c}}{2}\frac{\lambda }{\stackrel{˜}{c}},$ (3.5)

$\phi \left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\le \phi \left({x}_{\nu },{y}_{\nu }\right),$ (3.6)

${\text{argmin}}_{u,v}\left\{\phi \left(u,v\right)+\frac{2}{c+\stackrel{˜}{c}}h\left(u-\stackrel{˜}{x},v-\stackrel{˜}{y}\right)\right\}=\left\{\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\right\}.$ (3.7)

$‖\stackrel{˜}{y}-\stackrel{¯}{y}‖\le ‖\stackrel{˜}{y}-y‖+‖y-\stackrel{¯}{y}‖\le ‖{y}_{\nu }-y‖+‖y-\stackrel{¯}{y}‖\le ‖{y}_{\nu }-\stackrel{¯}{y}‖+2‖y-\stackrel{¯}{y}‖\le 3\eta .$ (3.8)

$‖\stackrel{˜}{x}-\stackrel{¯}{x}‖\le ‖\stackrel{˜}{x}-{x}_{\nu }‖+‖{x}_{\nu }-\stackrel{¯}{x}‖\le \left(2\stackrel{˜}{c}+1\right)\eta .$ (3.9)

$d\left({x}_{\nu },{G}^{-1}\left(y\right)\right)=d\left({x}_{\nu },{G}^{-1}\left(\stackrel{˜}{y}\right)\right)\le ‖\stackrel{˜}{x}-{x}_{\nu }‖<\lambda ,$

$\left(0,0\right)\in \partial \left(\psi +{\delta }_{gph\text{\hspace{0.17em}}G}\right)\left(\stackrel{˜}{x},\stackrel{˜}{y}\right).$ (3.10)

$\partial \psi \left(\stackrel{˜}{x},\stackrel{˜}{y}\right)=\frac{2}{c+\stackrel{˜}{c}}{\mathbb{B}}^{1}×\left(\frac{\stackrel{˜}{y}-y}{‖\stackrel{˜}{y}-y‖}+\frac{2\lambda }{c+\stackrel{˜}{c}}{\mathbb{B}}^{2}\right).$ (3.11)

$\left(0,0\right)\in \partial \psi \left(\stackrel{˜}{x},\stackrel{˜}{y}\right)+{N}_{gph\text{ }G}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right).$

$\left({x}_{0}^{*},-{y}_{0}^{*}\right)\in {\stackrel{^}{N}}_{gph\text{ }G}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right)\subset {N}_{gph\text{ }G}\left(\stackrel{˜}{x},\stackrel{˜}{y}\right),$

$w:=\frac{y-\stackrel{˜}{y}}{‖y-\stackrel{˜}{y}‖},\text{\hspace{0.17em}}\text{\hspace{0.17em}}‖w‖=1.$

$\begin{array}{c}〈{y}_{0}^{*},w〉-c‖{x}_{0}^{*}‖=〈-\frac{\stackrel{˜}{y}-y}{‖\stackrel{˜}{y}-y‖}-\frac{2\lambda }{c+\stackrel{˜}{c}}{z}_{2},w〉-\frac{2c}{c+\stackrel{˜}{c}}‖{z}_{1}‖\\ =1-\frac{2\lambda }{c+\stackrel{˜}{c}}〈{z}_{2},w〉-\frac{2c}{c+\stackrel{˜}{c}}‖{z}_{1}‖\\ \ge 1-\frac{2\lambda }{c+\stackrel{˜}{c}}-\frac{2c}{c+\stackrel{˜}{c}}\\ \ge 1-\frac{4\stackrel{˜}{c}\eta +2c}{c+\stackrel{˜}{c}}\\ >0,\end{array}$

$‖{x}_{\nu }-x‖\le \stackrel{˜}{c}‖{y}_{\nu }-y‖.$

(i) $gph\text{\hspace{0.17em}}{F}_{\stackrel{¯}{p}}$ 在点 $\left(\stackrel{¯}{x},0\right)$ 附近是局部闭的，

(ii) 存在 $\mathcal{l}>0$$0<\delta <\frac{1}{4}$ 使得

$F\left(p,x\right)\subseteq F\left(\stackrel{¯}{p},x\right)+\mathcal{l}d\left(p,\stackrel{¯}{p}\right)\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in {\mathbb{B}}_{\delta }\left(\stackrel{¯}{x}\right),\forall p\in {\mathbb{B}}_{\delta }\left(\stackrel{¯}{p}\right);$ (3.12)

(iii) ${‖{D}^{*}{F}_{\stackrel{¯}{p}}{\left(\stackrel{¯}{x}|\stackrel{¯}{y}\right)}^{-1}‖}^{+}

$S\left(p\right)\cap {\mathbb{B}}_{\delta }\left(\stackrel{¯}{x}\right)\subset S\left(\stackrel{¯}{p}\right)+\mathcal{l}\stackrel{˜}{c}d\left(p,\stackrel{¯}{p}\right)\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}d\left(p,\stackrel{¯}{p}\right)<\delta .$ (3.13)

$‖\stackrel{^}{x}-x‖\le \mathcal{l}\stackrel{˜}{c}d\left(p,\stackrel{¯}{p}\right).$ (3.14)

$‖y‖\le \mathcal{l}d\left(p,\stackrel{¯}{p}\right).$ (3.15)

$‖\stackrel{¯}{x}-x‖\le \stackrel{˜}{c}‖y-0‖.$ (3.16)

(i) $gph\text{\hspace{0.17em}}{F}_{\stackrel{¯}{p}}$ 在点 $\left(\stackrel{¯}{x},0\right)$ 附近是局部闭的；

(ii) 存在 $\mathcal{l}>0$ 以及 $\delta >0$ 使得

$F\left(p,x\right)\subseteq F\left(\stackrel{¯}{p},x\right)+\mathcal{l}d\left(p,\stackrel{¯}{p}\right)\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in {\mathbb{B}}_{\delta }\left(\stackrel{¯}{x}\right),\forall p\in {\mathbb{B}}_{\delta }\left(\stackrel{¯}{p}\right);$

(iii) ${F}_{\stackrel{¯}{p}}$ 在点 $\left(\stackrel{¯}{x},0\right)$ 处是度量正则的，正则系数为 $\kappa >reg\left({F}_{\stackrel{¯}{p}};\stackrel{¯}{x}|0\right)$

$S\left(p\right)=\left\{x\in {ℝ}^{n}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}0\in f\left(p,x\right)+H\left(p\right)\right\}.$ (3.17)

(a) f在点 $\left(\stackrel{¯}{p},\stackrel{¯}{x}\right)$ 处严格可微；

(b) H在点 $\stackrel{¯}{p}$ 处是局部上Lipschitz的。

$‖f\left(p,x\right)-f\left(\stackrel{˜}{p},\stackrel{˜}{x}\right)‖\le {\kappa }_{1}d\left(\left(p,x\right),\left(\stackrel{˜}{p},\stackrel{˜}{x}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall p,\stackrel{˜}{p}\in {U}_{p},\text{\hspace{0.17em}}\forall x,\stackrel{˜}{x}\in {U}_{x}$

$‖f\left(p,x\right)-f\left(\stackrel{¯}{p},x\right)‖\le {\kappa }_{1}d\left(p,\stackrel{˜}{p}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall p\in {U}_{p},\text{\hspace{0.17em}}\forall x\in {U}_{x}$ (3.18)

$f\left(p,x\right)\in f\left(\stackrel{¯}{p},x\right)+{\kappa }_{1}d\left(p,\stackrel{¯}{p}\right)\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall p\in {U}_{p},\text{\hspace{0.17em}}\forall x\in {U}_{x}.$ (3.19)

$H\left(p\right)\subset H\left(\stackrel{¯}{p}\right)+{\kappa }_{2}d\left(p,\stackrel{¯}{p}\right)\mathbb{B}.$ (3.20)

$f\left(p,x\right)+H\left(p\right)\subset f\left(\stackrel{¯}{p},x\right)+H\left(\stackrel{¯}{p}\right)+\left({\kappa }_{1}+{\kappa }_{2}\right)d\left(p,\stackrel{¯}{p}\right)\mathbb{B},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in {U}_{x},\forall p\in {U}_{p}\cap {\mathbb{B}}_{\gamma }\left(\stackrel{¯}{p}\right)$ ，即

$F\left(p,x\right)\subset F\left(\stackrel{¯}{p},x\right)+\left({\kappa }_{1}+{\kappa }_{2}\right)d\left(p,\stackrel{¯}{p}\right)\mathbb{B}.$ (3.21)

(i) $gph\text{\hspace{0.17em}}{F}_{\stackrel{¯}{p}}$ 在点 $\left(\stackrel{¯}{x},0\right)$ 附近是局部闭的；

(ii) ${‖{D}^{*}{F}_{\stackrel{¯}{p}}{\left(\stackrel{¯}{x}|\stackrel{¯}{y}\right)}^{-1}‖}^{+}

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