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 say that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$ if $$\mathcal{L}(\bar{x}, \bar{\mu}) = \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \bar{\mu}) \quad \text{and} \quad \mathcal{L}(\bar{x}, \bar{\mu}) = \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(\bar{x}, \mu).$$ We assume in this question that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$. a. Show that $\bar{x}$ is a solution of $(P)$: $\inf_{x \in K} f(x)$. b. Show that $\bar{\mu}$ is a solution of $(Q)$: $\sup_{\mu \in \mathbb{R}_+^p} G(\mu)$. c. Show that $\inf_{x \in \mathbb{R}^n} \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(x, \mu) = \sup_{\mu \in \mathbb{R}_+^p} \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$.
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 say that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$ if
$$\mathcal{L}(\bar{x}, \bar{\mu}) = \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \bar{\mu}) \quad \text{and} \quad \mathcal{L}(\bar{x}, \bar{\mu}) = \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(\bar{x}, \mu).$$
We assume in this question that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$.\\
a. Show that $\bar{x}$ is a solution of $(P)$: $\inf_{x \in K} f(x)$.\\
b. Show that $\bar{\mu}$ is a solution of $(Q)$: $\sup_{\mu \in \mathbb{R}_+^p} G(\mu)$.\\
c. Show that $\inf_{x \in \mathbb{R}^n} \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(x, \mu) = \sup_{\mu \in \mathbb{R}_+^p} \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$.