Noncompact infinite optimization and equilibria of constrained games in generalized convex spaces
The paper deals with the following infinite optimization problem: For a (finite or infinite) set \(I\), let \(X_i\) be topological spaces, \(i\in I\). Further, for \(i\in I\), let be \(X = \prod_{i\in I}X_i\), \(X^{-i} = \prod_{j\in I, j\neq i}X_j\), and let \(F_i:X^{-i} \rightarrow 2^{X_i}\) and \(f_i : X \rightarrow R\cup \{\pm \infty \}\) be a set-valued mapping and a function, respectively. The optimization problem is to find a solution \(\widehat{x}=(\widehat{x}_i, \widehat{x}^{-i}) \in X\) such that for each \(i\in I\), \(\widehat{x}_i \in F_i(\widehat{x}^{-i})\) and \(f_i(\widehat{x}_i) = \max f_i(y_i, \widehat{x}^{-i})\), where the maximum is taken over the set \(\{y_i \in F_i(\widehat{x}^{-i})\}\). The author obtains several existence theorems for the infinite optimization problem. They imply some results about the existence of equilibria in constrained \(n\)-person games. All the results are obtained under ''noncompact'' setting of generalized convex spaces without linear structure and generalize a number of known results to generalized convex spaces.
scientific article; zbMATH DE number 1570523