On the convergence of an active-set method for ℓ 1 minimization

Document Type

Article

Publication Date

12-1-2012

Abstract

We analyse an abridged version of the active-set algorithm FPC-AS proposed in Wen et al. [A fast algorithm for sparse reconstruction based on shrinkage, subspace optimisation and continuation, SIAM J. Sci. Comput. 32 (2010), pp. 1832-1857] for solving the l 1-regularized problem, i.e. a weighted sum of the l 1-norm |x| 1 and a smooth function f(x). The active-set algorithm alternatively iterates between two stages. In the first non-monotone line search (NMLS) stage, an iterative first-order method based on shrinkage is used to estimate the support at the solution. In the second subspace optimization stage, a smaller smooth problem is solved to recover the magnitudes of the non-zero components of x. We show that NMLS itself is globally convergent and the convergence rate is at least R-linearly. In particular, NMLS is able to identify the zero components of a stationary point after a finite number of steps under some mild conditions. The global convergence of FPC-AS is established based on the properties of NMLS. © 2012 Copyright Taylor and Francis Group, LLC.

Publication Source (Journal or Book title)

Optimization Methods and Software

First Page

1127

Last Page

1146

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