Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that
$$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$
is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by
$$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$
for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$.\\
(Uzawa's Theorem). Let $A \in \mathcal{M}_{p,n}(\mathbb{R})$ be a matrix of rank $p$ and $b \in \mathbb{R}^p$. We assume that the function $g$ is of the form
$$g : x \mapsto Ax + b.$$
a. Show that for all $\mu \in \mathbb{R}_+^p$, $\nabla f(x_\mu) = -{}^t A \mu$, and deduce that the function $\mu \mapsto x_\mu$ is continuous on $\mathbb{R}_+^p$.\\
b. Show that $(P)$ admits a unique solution $x^{\star} \in K$, and that $(Q)$ admits a unique solution $\mu^{\star} \in \mathbb{R}_+^p$.\\
Let $\rho > 0$. We define the sequence $(\mu^k)_{k \in \mathbb{N}}$ by recursion as follows:
\begin{itemize}
  \item we fix $\mu^0 \in \mathbb{R}_+^p$,
  \item for all $k \in \mathbb{N}$, we set $\mu^{k+1} = P_{\mathbb{R}_+^p}\left(\mu^k + \rho g(x_{\mu^k})\right)$,
\end{itemize}
where $P_{\mathbb{R}_+^p} : \mathbb{R}^p \rightarrow \mathbb{R}_+^p$ denotes the projection onto the closed convex set $\mathbb{R}_+^p$ of $\mathbb{R}^p$.\\
c. Show that $\mu^{\star} = P_{\mathbb{R}_+^p}\left(\mu^{\star} + \rho g(x_{\mu^{\star}})\right)$.\\
We now assume that $\|Ax\| \leqslant \sqrt{\frac{\alpha}{\rho}} \|x\|$ for all $x \in \mathbb{R}^n$.\\
d. Show that the sequence $(x_{\mu^k})_{k \in \mathbb{N}}$ converges to $x^{\star}$.\\
e. Show that the sequence $(\mu^k)_{k \in \mathbb{N}}$ converges to $\mu^{\star}$.