Iteratively reweighted ℓ1 algorithms with extrapolation

Iteratively reweighted ℓ1 algorithm is a popular algorithm for solving a large class of optimization problems whose objective is the sum of a Lipschitz differentiable loss function and a possibly nonconvex sparsity inducing regularizer. In this paper, motivated by the success of extrapolation techniques in accelerating first-order methods, we study how widely used extrapolation techniques such as those in [4,5,22,28] can be incorporated to possibly accelerate the iteratively reweighted ℓ1 algorithm. We consider three versions of such algorithms. For each version, we exhibit an explicitly checkable condition on the extrapolation parameters so that the sequence generated provably clusters at a stationary point of the optimization problem. We also investigate global convergence under additional Kurdyka-Lojasiewicz assumptions on certain potential functions. Our numerical experiments show that our algorithms usually outperform the general iterative shrinkage and thresholding algorithm in [21] and an adaptation of the iteratively reweighted ℓ1 algorithm in [23, Algorithm 7] with nonmonotone line-search for solving random instances of log penalty regularized least squares problems in terms of both CPU time and solution quality.