### Abstract

In this paper, we first demonstrate that positive semidefiniteness of a large well-structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked, in a linear fashion, to the matrix. We derive also the "dual counterpart" of the outlined representation, which expresses the possibility of positive semidefinite completion of a well-structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix. Using the representations, we then reformulate well-structured large-scale semidefinite problems into smooth convex-concave saddle point problems, which can be solved by a Prox-method developed in [6] with efficiency script O sign (ε^{-1}). Implementations and some numerical results for large-scale Lovász capacity and MAXCUT problems are finally presented.

Original language | English (US) |
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Pages (from-to) | 211-237 |

Number of pages | 27 |

Journal | Mathematical Programming |

Volume | 109 |

Issue number | 2-3 |

DOIs | |

State | Published - Mar 1 2007 |

Externally published | Yes |

### Keywords

- Mirror-Prox method
- Saddle point problem
- Semidefinite programming

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## Cite this

*Mathematical Programming*,

*109*(2-3), 211-237. https://doi.org/10.1007/s10107-006-0031-2