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For solving a wide class of nonconvex and nonsmooth problems, we propose aproximal linearized iteratively reweighted least squares (PL-IRLS) algorithm.We first approximate the original problem by smoothing methods, and secondwrite the approximated problem into an auxiliary problem by introducing newvariables. PL-IRLS is then built on solving the auxiliary problem by utilizingthe proximal linearization technique and the iteratively reweighted leastsquares (IRLS) method, and has remarkable computation advantages. We show thatPL-IRLS can be extended to solve more general nonconvex and nonsmooth problemsvia adjusting generalized parameters, and also to solve nonconvex and nonsmoothproblems with two or more blocks of variables. Theoretically, with the help ofthe Kurdyka- Lojasiewicz property, we prove that each bounded sequencegenerated by PL-IRLS globally converges to a critical point of the approximatedproblem. To the best of our knowledge, this is the first global convergenceresult of applying IRLS idea to solve nonconvex and nonsmooth problems. Atlast, we apply PL-IRLS to solve three representative nonconvex and nonsmoothproblems in sparse signal recovery and low-rank matrix recovery and obtain newglobally convergent algorithms.

Proximal linearized iteratively reweighted least squares for a class of nonconvex and nonsmooth problems