## Abstract

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming obtained from centrality equations of the form X S+SX=2 W, where W is a fixed positive definite matrix and 0 is a parameter, under the assumption that the problem has a strictly complementary primal-dual optimal solution. We present a different and simpler proof than the one given by Prei and Stoer [Prei, M. and Stoer, J., 2004, Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems. Mathematical Programming 99, 499-520.] that a weighted central path as a function of can be extended analytically beyond 0. In addition, the characterization of the limit points of the path and its normalized first-order derivatives is also provided. 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, we make some observation for the superlinear convergence of some primal-dual interior point SDP algorithms using AHO neighborhood.

Original language | English (US) |
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Pages (from-to) | 849-870 |

Number of pages | 22 |

Journal | Optimization Methods and Software |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2007 |

Externally published | Yes |

### Bibliographical note

Funding Information:The authors are in debt to three anonymous referees for invaluable advice and comments that have greatly improved this paper. This work was partially supported by NSF Grants CCR-9902010, CCR-0203113, INT-9910084 and CCF-0430644 and ONR grants N00014-03-1-0401 and N00014-05-1-0183.

## Keywords

- Error bound
- Limiting behavior
- Semidefinite programming
- Superlinear convergence
- Weighted central path