I have problems deriving formal proofs of following problem inspired in dynamic programming:
$V^{k+1}=\min_{\mu'}G(\mu',V^k)=T(V^k)$
$\mu^{k+1}=\arg \min_{\mu'} G(\mu',V^k)=\mu^*(V^k)$
where $V\in\mathcal{V}$ and $\mu\in\mathcal{U}$.
We make the following assumptions:
- $\mathcal{V}$ and $\mathcal{U}$ are compact.
- $G(\mu,V)$ is convex and differentiable in $\mu$ and thus $\mu^*(V)$ is unique.
- $\mu^*(V)$ is differentiable in $V$.
- $\mu^*(V)$ is an interior point for every $V\in \mathcal{V}$.
- $T$ is a contraction mapping, i.e., $\|\frac{\partial T(V)}{\partial V}\|<1$.
Since $T$ is a contraction mapping, $V^k$ converges to the (unique) fixed point of $T$. It is obvious then that $\mu^k$ also converges, since it is unique for each $V^k$ and $V^k$ converges.
However, I'm having problems deriving a "formal" proof for the whole fixed point system:
$(V,\mu)=F\left((V,\mu)\right)=(T(V),\mu^*(V))$
This fixed point system will have a unique solution if $F$ is a contraction mapping in the space $\mathcal{V}\times\mathcal{U}$, i.e., $\|\frac{\partial F(V,\mu)}{\partial (V,\mu)}\|<1$ in the product norm. Since $F$ does not depend on $\mu$, $\|\frac{\partial F(V,\mu)}{\partial \mu}\|=0$. Now choosing the $\infty$ norm for the Jacobian the norm would be:
$\|\frac{\partial F(V,\mu)}{\partial (V,\mu)}\|=\max\{\|\frac{\partial T(V)}{\partial V}\|,\|\frac{\partial \mu^*(V)}{\partial V}\|\}$
Now, from the assumptions we know that $\|\frac{\partial T(V)}{\partial V}\|<1$, but I don't know how to prove that $\|\frac{\partial \mu^*(V)}{\partial V}\|<1$ using the given assumptions, or if I overlooked something important.
Please feel free to correct any mistakes or give any ideas. Any help would be greatly appreciated.
Thank you very much.
I think you need regularity assumptions on the convexity with respect to $\mu$. If not a small change in $V$ may produce a large change in $\mu^*$. The mapping $T$ may well be Lipschitz-contracting without convexity in $\mu$. But usually $T$ is not differentiable without that convexity.