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 $g(z)\triangleq \inf _{w\in W}f(z,w)\ .$

$\forall w\in W,\forall z\in Z,g(z)\leq f(z,w)\$

$\Longrightarrow \forall w\in W,\sup _{z\in Z}g(z)\leq \sup _{z\in Z}f(z,w)\$

$\Longrightarrow \sup _{z\in Z}g(z)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\$

$\Longrightarrow \sup _{z\in Z}\inf _{w\in W}f(z,w)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\qquad \square \ .$

References

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

See also

• Minimax theorem