Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 205 / 6 / Solution

The term $\gamma \Vert \beta {\Vert }_2^2/2$ is strictly convex, while the remaining terms are convex, so the elastic net objective is strictly convex and its minimizer is unique. If two columns of $X$ are identical, swapping their coefficients leaves the objective unchanged. Uniqueness then forces those coefficients to be equal.

The Karush-Kuhn-Tucker conditions arewhereAssume $Y=X{\beta }^0$ and $\mathrm{sgn}(\hat{\beta })=\mathrm{sgn}({\beta }^0)=s$. The active equations giveThe inactive KKT inequalities becomeConversely, define ${\tilde{\beta }}_N=0$ andIf the displayed inequality holds and $\mathrm{sgn}({\tilde{\beta }}_S)=s_S$, the active equations and inactive inequalities together satisfy every KKT condition. Convexity and uniqueness imply $\tilde{\beta }=\hat{\beta }$, proving sign recovery.

The final sign condition printed in the question omits the factor $n$ before $\lambda$. For the objective as stated, the corrected expression above is required; without that correction, the claimed converse does not follow from the KKT equations.