## Abstract

In this paper we consider minimizing the spec tral condition number of a positive semidefinite matrix over a nonempty closed convex set Ω. We show that it can be solved as a convex programming problem, and moreover, the optimal value of the latter problem is achievable. As a consequence, when Ω is positive semidefinite representable, it can be cast into a semidefinite programming problem. We then propose a first-order method to solve the convex programming problem. The computational results show that our method is usually faster than the standard interior point solver SeDuMi [J. F. Sturm, Optim. Methods Softw., 11/12 (1999), pp. 625-653] while producing a comparable solution. We also study a closely related problem, that is, finding an optimal preconditioner for a positive definite matrix. This problem is not convex in general. We propose a convex relaxation for finding positive definite preconditioners. This relaxation turns out to be exact when finding optimal diagonal preconditioners.

Original language | English (US) |
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Pages (from-to) | 1193-1211 |

Number of pages | 19 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 32 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2011 |

Externally published | Yes |

## Keywords

- Condition number
- Convex programming
- Diagonal preconditioner
- Semidefinite programming