Elastic Net

The third and final installment in the Regularization series, resolving the instability the Lasso article deliberately left open. Derives the elastic net's naive and glmnet-style alpha/lambda parameterizations, proves from scratch that for identical standardized predictors any nonzero ridge component forces a unique equal split via strict convexity (verified analytically and by direct multi-seed simulation against lasso's genuinely arbitrary splits), and situates the general correlated case relative to the Zou-Hastie (2005) grouping-effect theorem. Works through the geometry of the resulting constraint region -- a curved-edged shape between Ridge's ball and Lasso's diamond that keeps the diamond's axis-aligned vertices (preserving sparsity) while replacing its flat edges with curvature (eliminating the coefficient discontinuities that gave Lasso trouble) -- and derives the coordinate-descent update in closed form, showing it algebraically contains both earlier articles' per-coordinate formulas as limiting cases. Two real worked examples, both direct callbacks to the Lasso article: the same 0.995-correlated Longley predictors (GNP and Year) that swapped discontinuously under lasso now split smoothly and near-equally the moment any ridge component is added; and the diabetes dataset's more tangled s1/s2 correlation, showing the same effect in a messier, multi-way-correlated real setting. Reports an honest, load-bearing complication rather than a tidy success story: a full cross-validated grid search over both alpha and lambda selects pure lasso (alpha = 1) as literally MSE-optimal on both datasets, meaning the grouping effect has to be chosen deliberately for interpretability and stability -- cross-validation alone does not surface it. Closes the trilogy with a synthesis of all three articles' 'no free lunch' throughline and a practical decision checklist for choosing among Ridge, Lasso, and Elastic Net.