Are minimax and maximin condition interchangable? I've came across a classic problem in my field where
$$\min_h \max_{\delta,\omega} |h^TP{(\delta,\omega)}-R_d(\delta,\omega)|$$
where $h$ ( a set of coefficients), $\omega$, and $\delta$ are independent variables.
My question here is that whether or not this 'minimax' problem can be rewritten/reformulated as a 'maximin' problem?
Such that
$$ \max_{\delta,\omega} \min_h |h^TP{(\delta,\omega)}-R_d(\delta,\omega)|$$
Will that make sense? Sorry if this is a silly question, but I tried googling it and wiki it. Doesn't really answer this general question!
Thank you in advance!
 A: There is an approach that addresses the problem in terms of interchange of limits on double sequences. The sufficient condition is the uniform convergence with respect to a parameter of the limits.
Fixed $ h $ exists a sequence $(\delta_k,\omega_k)\to (\delta^*,\omega^*)$ shout that 
$$
|h^TP{(\delta^{*},\omega^{*})}-R_d(\delta^*,\omega^*)|
=
\max_{\delta,\omega} |h^TP{(\delta,\omega)}-R_d(\delta,\omega)|.
$$
Now, there is a sequence $h_k\to h_*$ shout that 
$$
|h_*^TP{(\delta^*,\omega^*)}-R_d(\delta^*,\omega^*)|
=
\min_h \max_{\delta,\omega}|h^TP{(\delta,\omega)}-R_d(\delta,\omega)|$$ 
Let's $F_h(\delta,\omega)=|h^TP{(\delta,\omega)}-R_d(\delta,\omega)|$. Now note that
$$
\lim_{h\to h^*}\lim_{(\delta,\omega)\to(\delta^*,\omega^*)}F_h(\delta,\omega)=
\min_h  \max_{\delta,\omega} |h^TP{(\delta,\omega)}-R_d(\delta,\omega)|
$$
and 
$$
\lim_{(\delta,\omega)\to(\delta^*,\omega^*)}\lim_{h\to h^*}F_h(\delta,\omega)=
\max_{\delta,\omega} \min_h |h^TP{(\delta,\omega)}-R_d(\delta,\omega)|
$$
Then it applies the following theorem with $ t = h $ and $ x = (\delta, \omega) $

Theorem. Let $\{ F_t ; t\in T\}$ be a family of functions $F_t : X
 \rightarrow \mathbb{C}$ depending on a parameter t; let
  $\mathcal{B}_X$ be a base $X$ and $\mathcal{B}_{T}$ a base in $T$. If
  the family converges uniformly on $X$ over the base $\mathcal{B}_{T}$
  to a function $F : X \rightarrow \mathbb{C}$ and the limit
  $\lim_{\mathcal{B}_{T}} F_t(x)=A_t$ exists for each $t\in T$, the both
  repeated limits $\lim_{\mathcal{B}_{X}}(\lim_{\mathcal{B}_{T}}F_t(x))$
  and $\lim_{\mathcal{B}_{T}}(\lim_{\mathcal{B}_{X}}F_t(x))$ exist and
  the equality

Proof. See Zoric. P. 381. 
Within the limits of the above theorem replace $\lim_{\mathcal{B}_{T}}$ by $\lim_{h\to h^*}$ and $\lim_{\mathcal{B}_{X}}$ by $\lim_{(\delta,\omega)\to(\delta_*,\omega_*)}$.
