= KKT-consistent resolution of tied Lasso knots
{c}
{title2=$v=\operatorname*{argmin}\{\tfrac12v^TX^TXv-s_E^Tv_E:\ v_{E^c}=0,\ s_jv_j\geq0\ (j\in E\setminus N)\}$}
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 $X_j^TXv=s_j$ for $j\in N$ and for entering zero coordinates, and $s_jX_j^TXv\geq1$ 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
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