Tuning relation between Lasso and square-root Lasso ($\gamma =\lambda /{\hat{\sigma }}_{\lambda }$)

A Lasso minimizer with positive residual scale ${\hat{\sigma }}_{\lambda }=\Vert Y-X{\hat{\beta }}_{\lambda }{\Vert }_2/\sqrt{n}$ also minimizes the Square-root Lasso objective at the displayed tuning parameter. Divide its Karush-Kuhn-Tucker conditions by the residual scale and use the subgradient optimality condition.