In mathematics, the max–min inequality is as follows:
- For any function
![{\displaystyle \sup _{z\in Z}\inf _{w\in W}f(z,w)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84688425539d9663dbde9b9ceba59c589451652f)
When equality holds one says that f, W, and Z satisfies a strong max–min property (or a saddle-point property). The example function
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
References
See also