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 $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$. We assume throughout this question that the function $\mu \in \mathbb{R}_+^p \mapsto x_\mu$ is continuous. We consider a solution $\bar{\mu} \in \mathbb{R}_+^p$ of $(Q)$. a. Let $\mu \in \mathbb{R}_+^p$ and $\xi \in \mathbb{R}^p$ be such that $\mu + \xi \in \mathbb{R}_+^p$. Show that for all $t \in [0,1]$, $\mu + t\xi \in \mathbb{R}_+^p$, and $$\lim_{\substack{t \rightarrow 0 \\ t > 0}} \frac{G(\mu + t\xi) - G(\mu)}{t} = \langle g(x_\mu), \xi \rangle.$$ Deduce that for all $\mu \in \mathbb{R}_+^p$, $\langle g(x_{\bar{\mu}}), \mu - \bar{\mu} \rangle \leqslant 0$. b. Show that $x_{\bar{\mu}}$ is a solution of $(P)$.
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 $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$.\\
We assume throughout this question that the function $\mu \in \mathbb{R}_+^p \mapsto x_\mu$ is continuous. We consider a solution $\bar{\mu} \in \mathbb{R}_+^p$ of $(Q)$.\\
a. Let $\mu \in \mathbb{R}_+^p$ and $\xi \in \mathbb{R}^p$ be such that $\mu + \xi \in \mathbb{R}_+^p$. Show that for all $t \in [0,1]$, $\mu + t\xi \in \mathbb{R}_+^p$, and
$$\lim_{\substack{t \rightarrow 0 \\ t > 0}} \frac{G(\mu + t\xi) - G(\mu)}{t} = \langle g(x_\mu), \xi \rangle.$$
Deduce that for all $\mu \in \mathbb{R}_+^p$, $\langle g(x_{\bar{\mu}}), \mu - \bar{\mu} \rangle \leqslant 0$.\\
b. Show that $x_{\bar{\mu}}$ is a solution of $(P)$.