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Zbl 0960.90079
Lignola, M.Beatrice; Morgan, Jacqueline
Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. (English)
[J] J. Glob. Optim. 16, No.1, 57-67 (2000). ISSN 0925-5001; ISSN 1573-2916/e

In this paper, some notions of well-posedness are studied for parametric variational inequalities $VI(x)$ and for optimization problems with variational inequalities constraints $OPVIC$. The problem $VI(x)$ is defined by the pair $(A(x,u),K)$, where $A(x,.)$ is an operator from $E$ to $ E^{\ast }$ and $K\subset E$ is a nonempty closed convex set. The $OPVIC$ is intended as minimizing the function $f(x,u)$ over the set $\{(x,u)\in X\times K\mid u\in T(x)\}$, where $T(x)\subset E$ is the solution set of $VI(x)$. In both cases the variational inequalities considered are supposed to be uniquely solvable. \par The first notion studied is the parametrically strongly well-posedness of the family $VI(x)$, which is proven to be a generalization of the similar definition given by {\it T. Zolezzi} [Nonl. Anal., Theory Meth. Appl. 25, 437-453 (1995; Zbl 0841.49005)] for the case of parametric optimization problems. The authors give a characterization of the parametrically strongly well-posedness of $VI(x)$ for finite dimensional $E$ and a sufficient condition for the case $A(u)$ does not depend on $x$. For the latter case it is also given another characterization of the introduced concept in terms of the diameter of an $\epsilon $-solution set defined in a former paper. This last characterization can be extended only as a necessary condition to the general case $A(x,u)$. \par In a second section the authors introduce the concept of approximating sequences for $OPVIC$, which generalizes the same notion used in a former paper by the second author for bilevel programming problems. The notions of generalized and strongly well-posedness of $OPVIC$ are defined and sufficient conditions are provided. Both concepts are also characterized in case of finite dimensional $E$. Finally, an application of the introduced concepts to an exact penalty method is shortly presented.
[Walter Gomez Bofill (Cottbus)]

MSC 2000:: *90C30 Nonlinear programming
90C31 Sensitivity, etc.
58E35 Variational inequalities (global problems)

Keywords: optimization problems with variational inequalities constraints; parametric variational inequalities; well-posedness

Citations: Zbl 0841.49005

Cited in: Zbl 1080.49021

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