Penalty method problem
Consider the following problem
\begin{align} \text{min}&\quad \frac12(x^2_1+x^2_2)&& \\ \text{s.t}& \quad x_1+x_2=2&& \end{align}
a)Apply non-differentiable penalty $(i.e., f(x)+\rho \Vert h(x) \Vert _1 )$ and verify if there is a finite penalty parameter $(\rho>0)$ such that solution of the penalized problem coincides with the solution of the original problem.
observation: $f(x_1,x_2)=\frac12(x^2_1+x^2_2)$, $\quad h(x_1,x_2)=x_1+x_2 -2$ ,$\quad \Vert x \Vert _1=\sum_{i=1}^n \vert x_i \vert $
the solution of the original problem is (1,1)
I think we don't need the soccer here because by C-S $$\frac{1}{2}(x_1^2+x_2^2)=\frac{1}{4}(1^2+1^2)(x_1^2+x_2^2)\geq\frac{1}{4}(x_1+x_2)^2=1.$$ The equality occurs for $x_1=x_2=1$, which says that $1$ is a minimal value.