## Abstract

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.

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.

Original language | Undefined/Unknown |
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Pages (from-to) | 653-679 |

Number of pages | 27 |

Journal | Pure and applied functional analysis |

Volume | 3 |

Issue number | 4 |

State | Published - 2018 |