## Abstract

Certain matrix relationships play an important role in optimally conditions and algorithms for nonlinear and semidefinite programming. Let H be an n x n symmetric matrix. A an m x n matrix, and Z a basis for the null space of A. (In a typical optimization context H is the Hessian of a smooth function and A is the Jacobian of a set of constraints.) When the reduced Hessian Z^{T}HZ is positive definite, augmented Lagrangian methods rely on the known existence of a finite ρ̄ > O such that, for all ρ > ρ̄, the augmented Hessian H + ρA^{T}A is positive definite. In this note we analyze the case when Z^{T}HZ is positive semidefinite, i.e., singularity is allowed, and show that the situation is more complicated. In particular, we give a simple necessary and sufficient condition for the existence of a finite ρ̄ so that H + ρA^{T}A is positive semidefinite for ρ ≥ ρ̄. A corollary of our result is that if H is nonsingular and indefinite while Z^{T}HZ is positive semidefinite and singular, no such ρ̄ exists.

Original language | English (US) |
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Pages (from-to) | 243-253 |

Number of pages | 11 |

Journal | SIAM Journal on Optimization |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |

## Keywords

- Augmented Hessian
- Augmented Lagrangian methods
- Inertia
- Reduced Hessian

## ASJC Scopus subject areas

- Software
- Theoretical Computer Science