Necessary conditions for maxmin=minmax
The aim of the paper is to give necessary conditions for maximin=minimax. The following notions are necessary to formulate the main results: 1. \(\phi:\mathbb{R}^2\to \mathbb{R}\) is called a submaximum function if, for any \(u,v\in\mathbb{R}\), \(\phi(u,v)\leq uVv\). 2. Given a submaximum function \(\phi\), \(f\) is said to be \(\phi\)-concave-like if, for all \(\lambda>0\) and \(x_1,x_2\in X\), there exists \(x_3\in X\) such that \[ y\in Y\implies f(x_3,y)\geq \phi\left[f(x_1,y),f(x_2,y)\right]-\lambda \] where \(X\) and \(Y\) are nonempty sets and \(f:X\times Y\to \mathbb{R}\). 3. \(f\) is said to be weakly downward on \(X\) if, for all \(\varepsilon>0\), there exist \(0\leq\lambda<\varepsilon\), \(\delta>0\), such that, for all \(x_1,x_2\in X\), there exists \(x_3\in X\) such that \[ \begin{aligned} y\in Y &\implies f(x_3,y)+\lambda \geq f(x_1,y)\land f(x_2,y) \\ \text{and} | f(x_1,y)-f(x_2,y)| \geq \varepsilon &\implies f(x_3,y)\geq f(x_1,y) \land f(x_2,y)\delta. \end{aligned} \] The main results: a) If, (i) for any closed set \(K\subset Y\), \( \max_{x\in X} \min_{y\in K} f(x,y)= \min_{y\in K} \max_{x\in X} f(x,y); \) (ii) for any closed set \(K\subset Y\), the set of maximizers \( \arg \max_{x\in X} \{\min_{y\in K} f(x,y)\} \) is convex, then \(f\) is \(\phi\)-concave-like. b. If \(f\) is upward on \(Y\) and weakly downward on \(X\), then \[ \max_{x\in X} \min_{y\in Y} f(x,y)= \min_{y\in Y} \max_{x\in X} f(x,y). \] The paper contains other interesting results, as well.
scientific article; zbMATH DE number 1225570