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Zbl 0987.90077
Auslender, Alfred; Correa, Rafael
Primal and dual stability results for variational inequalities.
(English)
[J] Comput. Optim. Appl. 17, No.2-3, 117-130 (2000). ISSN 0926-6003; ISSN 1573-2894/e

It is considered the variational inequality problem (\text{VI}): find $\overline {x} \in C$ and $x^* \in A(\overline x)$ satisfying: $(x- \overline{x})^Tx^* \geq 0$ $\forall x \in C$, where $A:\bbfR^n\to 2^{(\bbfR^n)}$ is a maximal monotone operator, and $C= \{x \in \Bbb R^n \mid f_i(x) \leq 0$, $i=1 \dots n$, $Lx=l$ \}, with $f_i$ a closed proper convex function from $\bbfR^n$ to $\bbfR \bigcup {\infty},$ for each $i$ and $L-(q,n)$-matrix , $l \in \bbfR^q$. Other assumptions are established on $C$, one of them: $C \ne \emptyset$ and subset of $\operatorname {dom} A$ (domain of $A$). The solution set of (VI) is denoted by $S$. \par The authors investigate the continuous dependency of solutions of (VI) with respect to perturbations of the data in the following way: $A=A+ \varepsilon \widetilde{A}$, $f=f + \varepsilon \widetilde{f}$ , $L=L+ \varepsilon \widetilde{L}$ and $l=l+ \varepsilon \widetilde{l}$, where $\widetilde{A}$ is a maximal monotone operator, $\widetilde{f_i}$ are closed proper convex functions and $\widetilde{L}$ is a $(q,n)$-matrix, $\widetilde{l} \in \bbfR^n$. The set $C(\varepsilon)$, the perturbed variational inequality $(\text{VI}(\varepsilon)),$ its solution set $S(\varepsilon)$ and the multi-valued $S(.)$ are defined analogously. \par The concepts of $\text{VI}(\varepsilon)$-regular and $\text{VI}(0)$-stable, (given also in the paper), are central in the investigation. $\text{VI}(\varepsilon)$-regular is characterized by the generalized Slater condition and $S( \varepsilon) \ne \emptyset$, compact. Then the authors prove that if for all $\widetilde {A}, \widetilde{f_i}, i \dots m, \widetilde{L}, \widetilde{l}$ it is possible to find $\overline{\varepsilon}$ such that $\text{VI}(\varepsilon)$ is regular, for $0< \varepsilon< \overline{\varepsilon}$ then $\text{VI}(0)$ is said to be stable. Besides, they define the dual variational inequality problem ($D\text{VI}(\varepsilon)$) associated to $\text{VI}(\varepsilon)$ using its description as a generalized equation. Denoting $U(\varepsilon)$ as the solution's set of $D\text{VI}(\varepsilon)$, it is proven that: $\text{VI}(0)$-regular $\Rightarrow \text{VI}(0)$-stable, $S(.)$ and $U(.)$ locally stable at $0^+$. In addition: $\text{VI}(0)$ regular and $C$ subset of the interior of $\operatorname {dom} A$ implies $S(.)$ and $U(.)$ closed at $0^+$.
MSC 2000:
*90C31 Sensitivity, etc.

Keywords: variational inequalities; stability theory; perturbations; duality theory; recession functions

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