Throughout Part II, we consider a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ differentiable on $\mathbb{R}^n$ and $\alpha$-convex, where $\alpha \in \mathbb{R}_+^{\star}$. We set $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ where $p \in \mathbb{N}^{\star}$ and $g_1, \ldots, g_p$ are convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, differentiable on $\mathbb{R}^n$. We further assume that the set $K$ is non-empty. Show that there exists a unique element $x^{\star} \in K$ such that $f(x^{\star}) = \inf_{x \in K} f(x)$.
Throughout Part II, we consider a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ differentiable on $\mathbb{R}^n$ and $\alpha$-convex, where $\alpha \in \mathbb{R}_+^{\star}$. We set
$$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$
where $p \in \mathbb{N}^{\star}$ and $g_1, \ldots, g_p$ are convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, differentiable on $\mathbb{R}^n$. We further assume that the set $K$ is non-empty.\\
Show that there exists a unique element $x^{\star} \in K$ such that $f(x^{\star}) = \inf_{x \in K} f(x)$.