Full Length Paper

Mathematical Programming Computation

, Volume 3, Issue 3, pp 165-218

First online:

Templates for convex cone problems with applications to sparse signal recovery

  • Stephen R. BeckerAffiliated withApplied and Computational Mathematics, California Institute of Technology Email author 
  • , Emmanuel J. CandèsAffiliated withDepartments of Mathematics and of Statistics, Stanford University
  • , Michael C. GrantAffiliated withApplied and Computational Mathematics, California Institute of Technology

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This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||1 where W is arbitrary, or a combination thereof. In addition, the paper introduces a number of technical contributions such as a novel continuation scheme and a novel approach for controlling the step size, and applies results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.


Optimal first-order methods Nesterov’s accelerated descent algorithms Proximal algorithms Conic duality Smoothing by conjugation The Dantzig selector The LASSO Nuclear-norm minimization

Mathematics Subject Classification (2000)

90C05 90C06 90C25 62J077