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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!

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    See this question: http://math.stackexchange.com/questions/840299/is-minimax-equals-to-maximin2016-04-03

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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 $\lim_{(\delta,\omega)\to(\delta_*,\omega_*)}$.

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    If $R$ and $P$ are continuous try to prove that $ F(\delta, \omega) $ is a family of functions uniformimente equicontinuas with respect to the parameter $ h $ for the convertgencia uniform. If $R$ and $Q$ have continuous partial derivatives continuous attempt to prove, using the mean value theorem, which $F_h (\delta, \omega) $ is a family of functions uniformimente Lipchtiz with respect to the parameter $ h $. That is, the constant Lipchtiz does not depend on $ h $.2012-02-21