### 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) |
---|---|

Pages (from-to) | 243-253 |

Number of pages | 11 |

Journal | SIAM Journal on Optimization |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Optimization*,

*11*(1), 243-253. https://doi.org/10.1137/S1052623499351791

**A note on the augmented Hessian when the reduced Hessian is semidefinite.** / Anstreicher, Kurt M.; Wright, Margaret.

Research output: Contribution to journal › Article

*SIAM Journal on Optimization*, vol. 11, no. 1, pp. 243-253. https://doi.org/10.1137/S1052623499351791

}

TY - JOUR

T1 - A note on the augmented Hessian when the reduced Hessian is semidefinite

AU - Anstreicher, Kurt M.

AU - Wright, Margaret

PY - 2000

Y1 - 2000

N2 - 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 ZTHZ 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 + ρATA is positive definite. In this note we analyze the case when ZTHZ 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 + ρATA is positive semidefinite for ρ ≥ ρ̄. A corollary of our result is that if H is nonsingular and indefinite while ZTHZ is positive semidefinite and singular, no such ρ̄ exists.

AB - 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 ZTHZ 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 + ρATA is positive definite. In this note we analyze the case when ZTHZ 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 + ρATA is positive semidefinite for ρ ≥ ρ̄. A corollary of our result is that if H is nonsingular and indefinite while ZTHZ is positive semidefinite and singular, no such ρ̄ exists.

KW - Augmented Hessian

KW - Augmented Lagrangian methods

KW - Inertia

KW - Reduced Hessian

UR - http://www.scopus.com/inward/record.url?scp=0034549140&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034549140&partnerID=8YFLogxK

U2 - 10.1137/S1052623499351791

DO - 10.1137/S1052623499351791

M3 - Article

VL - 11

SP - 243

EP - 253

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 1

ER -