最大最小不等式

${\displaystyle g(z)\triangleq \inf _{w\in W}f(z,w)}$ と定義する。

${\displaystyle \implies g(z)\leq f(z,w),\forall z,w}$

${\displaystyle \implies \sup _{z}g(z)\leq \sup _{z}f(z,w),\forall w}$

${\displaystyle \implies \sup _{z}\inf _{w}f(z,w)\leq \sup _{z}f(z,w),\forall w}$

${\displaystyle \implies \sup _{z}\inf _{w}f(z,w)\leq \inf _{w}\sup _{z}f(z,w)\qquad \square }$

• 最小最大定理

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

• "Max Min of function less than Min max of function", http://math.stackexchange.com/q/186697/61602