A smoothing alternating minimization-based algorithm for clustering with sum-min of Euclidean norms

We consider the problem of minimizing an objective function defined as the finite sum of a minimum collection of nonsmooth
and convex functions, which includes the fundamental clustering problem as a particular case. To tackle this nonsmooth and nonconvex problem, we develop a smoothing alternating minimization-based algorithm
(SAMBA), which is proven to globally converge to a critical point of
the smoothed problem. We then show how it can be applied to the
clustering problem with adequate smoothing functions, producing two
very simple algorithms resembling the so-called k-means algorithm, with
global convergence analysis.