Convergence of Reweighted `1 Minimization Algorithms and Unique Solution of Truncated `p Minimization

Extensive numerical experiments have shown that the iteratively reweighted `1 minimization algorithm (IRL1) is a very efficient method for variable selection, signal reconstruction and image processing. However no convergence results have been given for the IRL1. In this paper, we first give a global convergence theorem of the IRL1 for the `2-`p (0 < p < 1) minimization problem. We prove that any sequence generated by the IRL1 converges to a stationary point of the `2-`p minimization problem. Moreover, the stationary point is a global minimizer in certain domain and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm. Other contribution of this paper is to prove the uniqueness of solution of the truncated `p minimization problem under the truncated null space property which is weaker than the restricted isometry property.