Show that in general $\inf \sup \ne \sup \inf$ for bilinear functions

Question

I am working on the following exercise:

Let $K \subseteq \mathbb{R}^n$, $L \subseteq \mathbb{R}^m$ and let $F(K,L) \rightarrow \mathbb{R}$ with

$$ F(x, \lambda) := c^Tx + \lambda^Tb - \lambda Ax \quad ( \ c \in \mathbb{R}^n, b \in \mathbb{R}^m, A \in \mathbb{R}^{m \times n}).$$

Let further

$$M := \inf_{x \in K} \sup_{\lambda \in L} \ F(x, \lambda)$$ $$N := \sup_{\lambda \in L} \inf_{x \in K} \ F(x, \lambda).$$

Show that in general $M \ne N$.

I can not find an example where that happens however. Could you help me?

Answer 1

How about $m=n=1$, $K=L=\{0,1\}$, $c=b=1$, $A=-2$. Then $F(x,\lambda) = x+\lambda -2x\lambda$.

We have:

• $\sup_\lambda F(0,\lambda)=1$
• $\sup_\lambda F(1,\lambda)=1$
• So $\inf_x\sup_\lambda F(x,\lambda)=1$

Whereas:

• $\inf_x F(x,0) = 0$
• $\inf_x F(x,1)=0$
• So $\sup_\lambda \inf_x F(x,\lambda)=0$.