Max–min inequality

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

For any function

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 For all , we get for all by definition of the infimum being a lower bound. Next, for all , for all by definition of the supremum being an upper bound. Thus, for all and , making an upper bound on for any choice of . Because the supremum is the least upper bound, holds for all . From this inequality, we also see that is a lower bound on . By the greatest lower bound property of infimum, . Putting all the pieces together, we get

which proves the desired inequality.



References

  • Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press.

See also

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.