Max–min inequality - Alchetron, The Free Social Encyclopedia

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

sup z ∈ Z inf w ∈ W f ( z , w ) ≤ inf w ∈ W sup z ∈ 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 does not always hold. 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 ) ≜ inf w ∈ W f ( z , w ) .

⟹ g ( z ) ≤ f ( z , w ) , ∀ z , w

⟹ sup z g ( z ) ≤ sup z f ( z , w ) , ∀ w

⟹ sup z inf w f ( z , w ) ≤ sup z f ( z , w ) , ∀ w

⟹ sup z inf w f ( z , w ) ≤ inf w sup z f ( z , w ) ◻

References

Max–min inequality Wikipedia

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