Journal article
A Note on the Augmented Hessian When the Reduced Hessian is Semidefinite
SIAM journal on optimization, Vol.11(1), pp.243-253
2000
DOI: 10.1137/S1052623499351791
Abstract
Certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming. Let H be an n × n symmetric matrix, A an m × 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 $\bar\rho\ge 0$ such that, for all $\rho > \bar\rho$, the augmented Hessian $H + \rho \ATA $ 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 $\bar\rho$ so that $H + \rho \ATA$ is positive semidefinite for $\rho \ge \bar\rho$. A corollary of our result is that if H is nonsingular and indefinite while ZTHZ is positive semidefinite and singular, no such $\bar\rho$ exists.
Details
- Title: Subtitle
- A Note on the Augmented Hessian When the Reduced Hessian is Semidefinite
- Creators
- Kurt M Anstreicher - University of IowaMargaret H Wright
- Resource Type
- Journal article
- Publication Details
- SIAM journal on optimization, Vol.11(1), pp.243-253
- Publisher
- Society for Industrial and Applied Mathematics
- DOI
- 10.1137/S1052623499351791
- ISSN
- 1052-6234
- eISSN
- 1095-7189
- Language
- English
- Date published
- 2000
- Academic Unit
- Industrial and Systems Engineering; Computer Science; Business Analytics
- Record Identifier
- 9984380543602771
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