Max–min inequality

In mathematics, the max–min inequality is as follows: for any function f: Z × W → ℝ,

$\sup_{z \in Z} \inf_{w \in W} f(z, w) \leq \inf_{w \in W} \sup_{z \in Z} f(z, w). \,$

When equality holds one says that f, W and Z satisfies a strong max–min property (or a saddle-point property). As the function f(z,w)=sin(z+w) illustrates, this equality not always holds. A theorem giving conditions on f, W and Z in order to guarantee the saddle point property is called a minimax theorem.

Proof

Define $g(z) \triangleq \inf_{w \in W} f(z, w)$.

$\Longrightarrow g(z) \leq f(z, w), \forall z, w$

$\Longrightarrow \sup_z g(z) \leq \sup_z f(z, w) , \forall w$

$\Longrightarrow \sup_z \inf_w f(z,w) \leq \sup_z f(z, w) \forall w$

$\Longrightarrow \sup_z \inf_w f(z,w) \leq \inf_w \sup_z f(z, w) \qquad \square$

See also

• Minimax theorem