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 $x \in \mathbb{R}^n$, and $x_k$ denotes the unique minimizer of $f_k$ on $\mathbb{R}^n$, and $x^\star$ the unique minimizer of $f$ on $K$. Show that the sequence $(f_k(x_k))_{k \in \mathbb{N}}$ converges to $f(x^{\star})$.
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 $x \in \mathbb{R}^n$, and $x_k$ denotes the unique minimizer of $f_k$ on $\mathbb{R}^n$, and $x^\star$ the unique minimizer of $f$ on $K$.\\
Show that the sequence $(f_k(x_k))_{k \in \mathbb{N}}$ converges to $f(x^{\star})$.