Convex Minimax Inequalities-Equalities

Abstract

The possibility of the interchange of the operations of sup and inf in a function of several variables is an important issue. For example, the main theorem of matrix game theory is the existence of a point (x ₀, y ₀) such that

$${x^T}A{y_0}x_0^TA{y_0}x_0^T{A_y}$$

for all x, y such that ${x_0}0,{y_i}0,\sum\limits_{i = 1}^n {{x_i} = 1}$, and $\sum\limits_{i = 1}^n {{y_i} = 1}$.