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The range condition asks whether a prescribed exact solution is also a minimizer of a penalized problem with suitably chosen artificial data. The subgradient optimality condition makes it equivalent to a source subgradient lying in the range of the adjoint forward operator.

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.