 Solution

An intercept-unpenalized Lasso estimator solves the constrained optimizationFor centered predictors, eliminate the intercept as in ridge regression and minimize $\Vert Y_c-X\beta {\Vert }_2^2$ under the same constraint. Equivalently, with a suitable tuning parameter $\lambda \ge 0$, use $\Vert Y-\alpha 1-X\beta {\Vert }_2^2+\lambda {\sum }_j|{\beta }_j|$. The constraint radius and penalty parameter are different parametrizations; larger radius permits less shrinkage.

The Lasso plot parametrizes the Lasso regularization path by the fraction of the maximum $l_1$ norm of the standardized slopes. Unlike the ridge quadratic penalty, the corners of the $l_1$ constraint can place some slopes exactly at zero, providing variable selection. The maximum norm refers to the path's unpenalized endpoint; centering and scaling conventions must match those used to construct that path.