A minimax theorem for vector-valued functions

Abstract

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

$$\alpha \varepsilon Max\bigcup\limits_{s\varepsilon X_o } {Min_w } f(s,Y_0 ),$$

there exists

$$\beta \varepsilon Min\bigcup\limits_{r\varepsilon Y_o } {Max} f(X_0 ,t),$$

such that β ≤ α (the exact meanings of the symbols are given in Section 2).