Error bounds and limiting behavior of weighted paths associated with the SDP MAP X 1/2SX 1/2

L. U. Zhaosong, Renato D.C. Monteiro

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Abstract

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X 1/2SX 1/2 = νW, where W is a fixed positive definite matrix and ν > 0 is a parameter, under the assumption that the problem has a strictly complementary primal-dual optimal solution. It is shown that a weighted central path as a function of √ν can be extended analytically beyond 0 and hence that the path converges as ν ↓. 0. Characterization of the limit points of the path and its normalized first-order derivatives are also provided. As a consequence, it is shown that a weighted central path can have two types of behavior: it converges either as θ(ν) or as θ(√ν) depending on whether the matrix W on a certain scaled space is block diagonal or not, respectively. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal-dual optimal solutions. Finally, in light of the results of this paper, we give a characterization of a sufficient condition proposed by Potra and Sheng which guarantees the superlinear convergence of a class of primal-dual interior-point SDP algorithms.

Original languageEnglish (US)
Pages (from-to)348-374
Number of pages27
JournalSIAM Journal on Optimization
Volume15
Issue number2
DOIs
StatePublished - May 27 2005
Externally publishedYes

Keywords

  • Error bound
  • Limiting behavior
  • Semideflnite programming (SDP)
  • Superlinear convergence
  • Weighted central path

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