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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.

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