Suppose $\widehat\beta_1$ and $\widehat\beta_2$ minimize $Q$. Their midpoint is also a minimizer because $Q$ is a <convex function>. The group penalty is convex, while the squared Euclidean norm is strictly convex in the fitted value. If $X\widehat\beta_1\ne X\widehat\beta_2$, strict convexity would make the midpoint objective strictly smaller than the minimum. Hence $X\widehat\beta_1=X\widehat\beta_2$, so the fitted values are unique.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 1 / d / Solution

 Solution (source code)