Abstract
Recently, Xiao et al. proposed a nonsmooth equations‐based method to solve the ℓ1‐norm minimization problem (2011). The advantage of this method is its simplicity and lower storage. In this paper, based on new nonsmooth equations reformulation, we investigate new nonsmooth equations‐based algorithms for solving ℓ1‐norm minimization problems. Under mild conditions, we show that the proposed algorithms are globally convergent. The preliminary numerical results demonstrate the effectiveness of the proposed algorithms.
Highlights
We consider the 1-norm minimization problem minf x x1.1 where x ∈ Rn, b ∈ Rm, A ∈ Rm×n, and ρ is a nonnegative parameter
The convex optimization problem 1.1 can be cast as a second-order cone programming problem and could be solved via interior point methods
One of the most popular algorithms falls into the iterative shrinkage/thresholding IST class 6, 7
Summary
|vi|2 and v1 i |vi| to denote the Euclidean norm and the 1-norm of vector v ∈ Rn, respectively. In many applications, the problem is large scale and involves dense matrix data, which often precludes the use and potential advantage of sophisticated interior point methods This motivated the search of simpler first-order algorithms for solving 1.1 , where the dominant computational effort is a relatively cheap matrix-vector multiplication involving A and AT. Xiao et al 17 developed a nonsmooth equations-based algorithm called SGCS for solving 1-norm minimization problems in CS They reformulated the box-constrained quadratic program obtained by Figueiredo et al 12 into a system of nonsmooth equations and applied the spectral gradient projection method 18 to solving the nonsmooth equation. Throughout the paper, we use · , · to denote the inner product of two vectors in Rn
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