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In this article, we propose an algorithm, NESTA-LASSO, for the LASSO problem,i.e., an underdetermined linear least-squares problem with a 1-norm constrainton the solution. We prove under the assumption of the restricted isometryproperty (RIP) and a sparsity condition on the solution, that NESTA-LASSO isguaranteed to be almost always locally linearly convergent. As in the case ofthe algorithm NESTA proposed by Becker, Bobin, and Candes, we rely onNesterov's accelerated proximal gradient method, which takes O(e^{-1/2})iterations to come within e > 0 of the optimal value. We introduce amodification to Nesterov's method that regularly updates the prox-center in aprovably optimal manner, and the aforementioned linear convergence is in partdue to this modification.  In the second part of this article, we attempt to solve the basis pursuitdenoising BPDN problem (i.e., approximating the minimum 1-norm solution to anunderdetermined least squares problem) by using NESTA-LASSO in conjunction withthe Pareto root-finding method employed by van den Berg and Friedlander intheir SPGL1 solver. The resulting algorithm is called PARNES. We providenumerical evidence to show that it is comparable to currently availablesolvers.

PARNES: A rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signals