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Max–min inequality

Mathematical 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

References

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mathematical-optimization inequalities-(mathematics)

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