### Abstract

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X ^{1/2}SX ^{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 language | English (US) |
---|---|

Pages (from-to) | 348-374 |

Number of pages | 27 |

Journal | SIAM Journal on Optimization |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - May 27 2005 |

Externally published | Yes |

### Keywords

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

## Fingerprint Dive into the research topics of 'Error bounds and limiting behavior of weighted paths associated with the SDP MAP X <sup>1/2</sup>SX <sup>1/2</sup>'. Together they form a unique fingerprint.

## Cite this

^{1/2}SX

^{1/2}

*SIAM Journal on Optimization*,

*15*(2), 348-374. https://doi.org/10.1137/S1052623403430828