I am trying to learn continuous optimization and I need to solve the following exercise. Despite the fact that I solved some exercises about KKT conditions and Lagrange multipliers, I can't solve this one. It seems to require an upper level of rigor and formalism. That's the exercise:
Given matrix $ A \in \mathbb R^{n \times m}$ and vector $ b \in \mathbb R^{m}$, Let $f$ be $$ f(x) := \|Ax +b \|_1^2 + \alpha \|x\|_1^2$$
Show that for every number $\alpha > 0$ function $f$ has exactly one unconstrained global miniminum at $x_\alpha$.
Moreover, show that each $x_\alpha$ is a KKT point for problem of minimizing $g(x) = \|Ax+b\|_1^2$ with $x$ in an Euclidean ball centered at zero with radius $\Delta_\alpha$.
Finally, show the associated Lagrange multipliers.
How do you solve these? Thanks in advance.