Journal of Optimization Theory and Applications

, Volume 72, Issue 1, pp 7–35

On the convergence of the coordinate descent method for convex differentiable minimization

  • Z. Q. Luo
  • P. Tseng
Contributed Papers

DOI: 10.1007/BF00939948

Cite this article as:
Luo, Z.Q. & Tseng, P. J Optim Theory Appl (1992) 72: 7. doi:10.1007/BF00939948

Abstract

The coordinate descent method enjoys a long history in convex differentiable minimization. Surprisingly, very little is known about the convergence of the iterates generated by this method. Convergence typically requires restrictive assumptions such as that the cost function has bounded level sets and is in some sense strictly convex. In a recent work, Luo and Tseng showed that the iterates are convergent for the symmetric monotone linear complementarity problem, for which the cost function is convex quadratic, but not necessarily strictly convex, and does not necessarily have bounded level sets. In this paper, we extend these results to problems for which the cost function is the composition of an affine mapping with a strictly convex function which is twice differentiable in its effective domain. In addition, we show that the convergence is at least linear. As a consequence of this result, we obtain, for the first time, that the dual iterates generated by a number of existing methods for matrix balancing and entropy optimization are linearly convergent.

Key Words

Coordinate descentconvex differentiable optimizationsymmetric linear complementarity problems

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Z. Q. Luo
    • 1
  • P. Tseng
    • 2
  1. 1.Communications Research Laboratory, Department of Electrical and Computer EngineeringMcMaster UniversityHamiltonCanada
  2. 2.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridge
  3. 3.Department of MathematicsUniversity of WashingtonSeattle