Mathematical Programming Computation

, 3:165

Templates for convex cone problems with applications to sparse signal recovery

Authors

    • Applied and Computational MathematicsCalifornia Institute of Technology
  • Emmanuel J. Candès
    • Departments of Mathematics and of StatisticsStanford University
  • Michael C. Grant
    • Applied and Computational MathematicsCalifornia Institute of Technology
Full Length Paper

DOI: 10.1007/s12532-011-0029-5

Cite this article as:
Becker, S.R., Candès, E.J. & Grant, M.C. Math. Prog. Comp. (2011) 3: 165. doi:10.1007/s12532-011-0029-5

Abstract

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.

Keywords

Optimal first-order methodsNesterov’s accelerated descent algorithmsProximal algorithmsConic dualitySmoothing by conjugationThe Dantzig selectorThe LASSONuclear-norm minimization

Mathematics Subject Classification (2000)

90C0590C0690C2562J077

Copyright information

© Springer and Mathematical Optimization Society 2011