For all $k \in \mathbb{N}$, we introduce the function $f_k : \mathbb{R}^n \rightarrow \mathbb{R}$ defined by $$f_k(x) = f(x) + k\Psi(x) \text{ for all } x \in \mathbb{R}^n$$ where $\Psi : \mathbb{R}^n \rightarrow \mathbb{R}$ is the function defined by $\Psi(x) = \sum_{i=1}^{p} \max(0, g_i(x))^2$ for all $x \in \mathbb{R}^n$. Show that for all $k \in \mathbb{N}$, there exists a unique $x_k \in \mathbb{R}^n$ such that $f_k(x_k) = \inf_{x \in \mathbb{R}^n} f_k(x)$. Hint: one may begin by showing that if $g : \mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, and $h : \mathbb{R} \rightarrow \mathbb{R}$ is a convex increasing function, then $h \circ g$ is convex.
For all $k \in \mathbb{N}$, we introduce the function $f_k : \mathbb{R}^n \rightarrow \mathbb{R}$ defined by
$$f_k(x) = f(x) + k\Psi(x) \text{ for all } x \in \mathbb{R}^n$$
where $\Psi : \mathbb{R}^n \rightarrow \mathbb{R}$ is the function defined by $\Psi(x) = \sum_{i=1}^{p} \max(0, g_i(x))^2$ for all $x \in \mathbb{R}^n$.\\
Show that for all $k \in \mathbb{N}$, there exists a unique $x_k \in \mathbb{R}^n$ such that $f_k(x_k) = \inf_{x \in \mathbb{R}^n} f_k(x)$.\\
Hint: one may begin by showing that if $g : \mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, and $h : \mathbb{R} \rightarrow \mathbb{R}$ is a convex increasing function, then $h \circ g$ is convex.