  A second-order block coordinate descent method is proposed for the
unconstrained minimization of an objective function with Lipschitz continuous
Hessian. At each iteration, a block of variables is selected by means of a
greedy (Gauss-Southwell) rule which considers the amount of first-order
stationarity violation, then an approximate minimizer of a cubic model is
computed for the block update. In the proposed scheme, blocks are not required
to have a prefixed structure and their size is allowed to change during the
iterations. For non-convex objective functions, global convergence to
stationary points is proved and a worst-case iteration complexity analysis is
provided. In particular, given a tolerance $\varepsilon$, we show that at most
${\cal O}(\varepsilon^{-3/2})$ iterations are needed to drive the stationarity
violation with respect to the selected block of variables below $\varepsilon$,
while at most ${\cal O}(\varepsilon^{-2})$ iterations are needed to drive the
stationarity violation with respect to all variables below $\varepsilon$.
Numerical results are finally provided.
