A first-order smoothed penalty method for compressed sensing

N. S. Aybat, G. Iyengar

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


We propose a first-order smoothed penalty algorithm (SPA) to solve the sparse recovery problem min{||x||1: Ax = b}. SPA is efficient as long as the matrix-vector product Ax and AT y can be computed efficiently; in particular, A need not have orthogonal rows. SPA converges to the target signal by solving a sequence of penalized optimization subproblems, and each subproblem is solved using Nesterov's optimal algorithm for simple sets [Yu. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer Academic Publishers, Norwell, MA, 2004] and [Yu. Nesterov, Math. Program., 103 (2005), pp. 127-152]. We show that the SPA iterates xk are ∈-feasible; i.e. || Axk - b ||2 ≤ e and e-optimal; i.e. | ||xk ||1 - ||x*||1| ≤ ∈ after Õ (∈-2/3) iterations. SPA is able to work with l 1, l2, or l penalty on the infeasibility, and SPA can be easily extended to solve the relaxed recovery problem min{||x||1: ||Ax - b||2 ≤θ}.

Original languageEnglish (US)
Pages (from-to)287-313
Number of pages27
JournalSIAM Journal on Optimization
Issue number1
StatePublished - 2011

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Applied Mathematics


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