Max–min inequality

In mathematics, the max–min inequality is as follows:


 * For any function $$\ f : Z \times W \to \mathbb{R}\ ,$$

\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). The example function $$\ f(z,w) = \sin( z + w )\ $$ illustrates that the equality does not hold for every function.

A theorem giving conditions on $f$, $W$, and $Z$ which guarantee the saddle point property is called a minimax theorem.

Proof
Define $$ g(z) \triangleq \inf_{w \in W} f(z, w)\ .$$ For all $$z \in Z$$, we get $g(z) \leq f(z, w) $  for all $$w \in W $$ by definition of the infimum being a lower bound. Next, for all $w \in W $, $$f(z, w) \leq \sup_{z \in Z} f(z, w) $$ for all $z \in Z $ by definition of the supremum being an upper bound. Thus, for all $$z \in Z $$ and $$w \in W $$, $$g(z) \leq f(z, w) \leq \sup_{z \in Z} f(z, w) $$ making $$h(w) \triangleq \sup_{z \in Z} f(z, w) $$ an upper bound on $$g(z) $$ for any choice of $$w \in W $$. Because the supremum is the least upper bound, $$\sup_{z \in Z} g(z) \leq h(w) $$ holds for all $$w \in W $$. From this inequality, we also see that $$\sup_{z \in Z} g(z) $$ is a lower bound on $$h(w) $$. By the greatest lower bound property of infimum, $$\sup_{z \in Z} g(z) \leq \inf_{w \in W} h(w) $$. Putting all the pieces together, we get

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

which proves the desired inequality. $$\blacksquare $$