Let $LaTeX: \textstyle f : \mbox{ X } \times \mbox{ Y } \to \mathbb{R}.$ Then, $LaTeX: \textstyle (x^*, y^*)$ is a saddle point of $LaTeX: \textstyle f$ if $LaTeX: \textstyle x^*$ minimizes $LaTeX: \textstyle f(x, y^*)$ on $LaTeX: \textstyle \mbox{ X }$, and $LaTeX: \textstyle y^*$ maximizes $LaTeX: \textstyle f(x^*, y)$ on $LaTeX: \textstyle \mbox{ Y }$. Equivalently,

$LaTeX: f(x^*, y) \le f(x^*, y^*) \le f(x, y^*) \mbox{ for all } x \mbox{ in X, } y \mbox{ in Y.}$

Von Neumann (1928) proved this equivalent to:

$LaTeX: \begin{array}{rcl} f(x^*, y^*) & = & \inf \{ \sup \{ f(x, y) : y \in \mbox{ Y } \} : x \in \mbox{ X } \} \\ & = & \sup \{ \inf \{ f(x, y) : x \in \mbox{ X } \} : y \in \mbox{ Y } \}. \end{array}$

A sufficient condition for a saddle point to exist is that $LaTeX: \textstyle \mbox{ X }$ and $LaTeX: \textstyle \mbox{ Y }$ are non-empty, compact, convex sets, $LaTeX: f(.,y)$ is convex on $LaTeX: \textstyle \mbox{ X }$ for each $LaTeX: y$ in $LaTeX: \textstyle \mbox{ Y }$, and $LaTeX: f(x,.)$ is concave on $LaTeX: \textstyle \mbox{ Y }$ for each $LaTeX: x$ in $LaTeX: \textstyle \mbox{ X }$.