Results 1  10
of
12
V.: Spectral analysis of saddle point matrices with indefinite leading blocks
 SIAM J. Matrix Anal. Appl
, 2010
"... Abstract. We provide eigenvalue intervals for symmetric saddlepoint and regularised saddlepoint matrices in the case where the (1,1) block may be indefinite. These generalise known results for the definite (1,1) case. We also study the spectral properties of the equivalent augmented formulation, w ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We provide eigenvalue intervals for symmetric saddlepoint and regularised saddlepoint matrices in the case where the (1,1) block may be indefinite. These generalise known results for the definite (1,1) case. We also study the spectral properties of the equivalent augmented formulation, which is an alternative to explicitly dealing with the indefinite (1,1) block. Such an analysis may be used to assess the convergence of suitable Krylov subspace methods. We conclude with spectral
A note on the augmented hessian when the reduced hessian is semidefinite. Working paper
, 1999
"... 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 sm ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
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 ¯ρ ≥ 0 such that, for all ρ>¯ρ, 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 ¯ρ so that H + ρATA is positive semidefinite for ρ ≥ ¯ρ. A corollary of our result is that if H is nonsingular and indefinite while ZTHZ is positive semidefinite and singular, no such ¯ρ exists.
Mol l in, Quadratics
, 1996
"... iterative workingset method for largescale nonconvex ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
iterative workingset method for largescale nonconvex
Survey Paper A Fresh VariationalAnalysis Look at the Positive Semidefinite Matrices World
, 2011
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract
 Add to MetaCart
(Show Context)
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Contents
"... More than you wanted to know about quadratic forms KC Border v. 2012.08.24::15.59 ..."
Abstract
 Add to MetaCart
More than you wanted to know about quadratic forms KC Border v. 2012.08.24::15.59
unknown title
"... Abstract. We provide eigenvalue intervals for symmetric saddle point and regularized saddle point matrices in the case where the (1,1) block may be indefinite. These generalize known results for the definite (1,1) case. We also study the spectral properties of the equivalent augmented formulation, w ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We provide eigenvalue intervals for symmetric saddle point and regularized saddle point matrices in the case where the (1,1) block may be indefinite. These generalize known results for the definite (1,1) case. We also study the spectral properties of the equivalent augmented formulation, which is an alternative to explicitly dealing with the indefinite (1,1) block. Such an analysis may be used to assess the convergence of suitable Krylov subspace methods. We conclude with spectral
(Revised)
, 2009
"... On solving trustregion and other regularised subproblems in optimization c © Science and Technology Facilities Council Enquires about copyright, reproduction and requests for additional copies of this report should be addressed to: ..."
Abstract
 Add to MetaCart
(Show Context)
On solving trustregion and other regularised subproblems in optimization c © Science and Technology Facilities Council Enquires about copyright, reproduction and requests for additional copies of this report should be addressed to: