 KKT-consistent resolution of tied Lasso knots

At a positive penalty level, let $E$ be the residual-score equality set, $s_j$ its signs and $N$ the nonzero regression coefficient support. With full column rank, the displayed strictly convex quadratic program selects the direction for a small decrease of the penalty. Existing nonzero regression coefficients have unrestricted tangent components; tied zero regression coefficients may move only with their residual signs. Stationarity gives $X_j^{T}Xv=s_j$ for $j\in N$ and for entering zero coordinates, and $s_j{X}_j^{T}Xv\ge 1$ for tied coordinates that remain zero. These are exactly the first-order Karush-Kuhn-Tucker conditions for the Lasso along the next segment. Normalize $Xv$ to obtain the corresponding least-angle direction. This resolves simultaneous entries or dropouts without adding a generic-position assumption.

// Target: mathematical-optimization.bigb