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A minimax theorem for vector-valued functions. (English) Zbl 0631.90077

In this work, as usual in vector-valued optimization, we consider the partial ordering induced in a topological vector space by a closed and convex cone. In this way, we define maximal and minimal sets of a vector- valued function and consider minimax problems in this setting. Under suitable hypotheses (continuity, compactness, and special types of convexity), we prove that, for every

$\alpha \in Max{\cup }_{s\in {X}_{0}}Mi{n}_{w}f\left(s,{Y}_{0}\right),$

there exists

$\beta \in Min{\cup }_{t\in {Y}_{0}}Maxf\left({X}_{0},t\right)$

such that $\beta \le \alpha$ (the exact meanings of the symbols are given in Section 2).

##### MSC:
 90C31 Sensitivity, stability, parametric optimization
##### Keywords:
vector-valued optimization; minimax problems
##### References:
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