= Solution
<Coordinate descent> cycles through the intercept and coefficient coordinates, minimizing the convex ridge objective in one coordinate while holding the others fixed. Given current coefficients, update
$$
\alpha\leftarrow\frac1n\sum_{i=1}^n
\left(Y_i-\sum_{k=1}^pX_{ik}\beta_k\right).
$$
For coordinate $j$, form the partial residual
$$
r^{(j)}=Y-\alpha\mathbf1-\sum_{k\ne j}X_k\beta_k
$$
and update it by the exact one-dimensional minimizer
$$
\beta_j\leftarrow
\frac{X_j^Tr^{(j)}}{X_j^TX_j+\lambda}.
$$
Repeated sweeps converge to the unique fitted value because the objective is a <convex function>; with $\lambda>0$ it is strictly convex in $\beta$.
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