Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. (English) Zbl 1191.90079

Li, M. H.; Li, S. J.; Zhang, W. Y.

J. Ind. Manag. Optim. 5, No. 4, 683-696 (2009).

Considered are two classes of generalized vector quasi-equilibrium problem of finding an \(x^0\in E\) such that
\[ x^0\in K(x^0) \quad \text{and}\quad F(x^0,y)\not\subset -\operatorname{int}C (x^0),\quad \forall y\in K(x^0), \]
and of finding an \(x^1\in E\) such that
\[ x^1\in K(x^1) \quad \text{and}\quad F(x^1,y)\subset -C(x^1),\quad \forall y\in K(x^1). \]
where \((X,d) \) is a metric space, \(Z\) is a locally convex Haussdorff topological space, \(E\) is a nonempty closed and convex subset in \( X,C:E\rightrightarrows Z\) is a set-valued mapping such that for any \(x\in E\), \(C(x)\) is a proper, pointed, closed and convex cone in \(Z\) with nonempty interior \(\operatorname{int}C(x)\), \(K:E\rightrightarrows E\) and \( F:E\times E\rightrightarrows Z\) are two nontrivial (i.e., their images are nonempty for all \(x\in E\)) set-valued maps.
The authors introduce the definitions of the Levitin-Polyak well-posedness for these problems and their criteria and characterizations are investigated.
Some equivalent relations between the Levitin-Polyak well-posedness for optimization problems and the Levitin-Polyak well-posedness for generalized vector quasi-equilibrium problems are obtained. Finally, the authors derive a set-valued version of Ekeland’s variational principle and apply it to establish a sufficient condition for the Levitin-Polyak well-posedness of a class of generalized vector quasi-equilibrium problems.

Reviewer: I. M. Stancu-Minasian (Bucureşti)

MSC:

90C31	Sensitivity, stability, parametric optimization
49J53	Set-valued and variational analysis

Keywords:

Ekeland’s variational principle