The two <Lasso> problems are
$$
\min_\beta\lVert Y-x\beta\rVert_2^2\quad\text{subject to }|\beta|\leq t
$$
and
$$
\min_{\beta_1,\beta_2}\lVert Y-x(\beta_1+\beta_2)\rVert_2^2
\quad\text{subject to }|\beta_1|+|\beta_2|\leq t.
$$
If $\widehat\beta$ solves the first problem, the complete solution set of the second is
$$
\boxed{\{(\beta_1,\beta_2):\beta_1+\beta_2=\widehat\beta,\ 
|\beta_1|+|\beta_2|\leq t\}}.
$$
Indeed, the feasible totals are exactly $[-t,t]$, the same as in the first problem.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-218/1/c/solution