• Corpus ID: 239885291

Eigenvalue Bounds for Double Saddle-Point Systems

  title={Eigenvalue Bounds for Double Saddle-Point Systems},
  author={Susanne Bradley and Chen Greif},
We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from… 

Figures from this paper

A Preconditioned Inexact Active-Set Method for Large-Scale Nonlinear Optimal Control Problems
A global convergence proof of the recently proposed sequential homotopy method with an inexact Krylov–semismooth-Newton method employed as a local solver and an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach is provided.


Spectral Analysis of Saddle Point Matrices with Indefinite Leading Blocks
Eigenvalue intervals for symmetric saddle point and regularized saddle point matrices in the case where the (1,1) block may be indefinite are provided and the spectral properties of the equivalent augmented formulation are studied.
A Note On Symmetric Positive Definite Preconditioners for Multiple Saddle-Point Systems
A preconditioner is described for multiple saddle-point systems of block tridiagonal form which can be applied within the Minres algorithm, and which has only two distinct eigenvalues, 1 and −1, when the preconditionser is applied exactly.
Refining the Lower Bound on the Positive Eigenvalues of Saddle Point Matrices with Insights on the Interactions between the Blocks
This paper illustrates how and in which circumstances the convergence of Minres might be affected by these few very small eigenvalues in the (1,1) block, and derives theoretically a tighter lower bound on the positive eigen values of saddle point matrices of the KKT form.
Natural Preconditioning and Iterative Methods for Saddle Point Systems
This survey concerns iterative solution methods for quadratic extremum problems and shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods.
Spectral estimates for unreduced symmetric KKT systems arising from Interior Point methods
New spectral estimates are presented for symmetrized Karush–Kuhn–Tucker systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods: the new bounds are established for the unpreconditioned matrices and for the matrices preconditionsed by symmetric positive definite augmented preconditioners.
Numerical solution of saddle point problems
A large selection of solution methods for linear systems in saddle point form are presented, with an emphasis on iterative methods for large and sparse problems.
A Note on Preconditioning for Indefinite Linear Systems
For nonsingular indefinite matrices of saddle-point (or KKT) form, it is shown how preconditioners incorporating an exact Schur complement lead to preconditionsed matrices with exactly two or exactly three distinct eigenvalues.
Robust Preconditioners for Multiple Saddle Point Problems and Applications to Optimal Control Problems
A characterization of all block structured norms which ensure well-posedness of multiple saddle point problems with block tridiagonal Hessian in a Hilbert space setting is given and the generality of this approach is demonstrated with two optimal control problems related to the heat and the wave equation.
Bounds on Eigenvalues of Matrices Arising from Interior-Point Methods
Interior-point methods feature prominently among numerical methods for inequality-constrained optimization problems, and involve the need to solve a sequence of linear systems that typically become...
Iterative Methods for Double Saddle Point Systems
  • F. Beik, M. Benzi
  • Mathematics, Computer Science
    SIAM J. Matrix Anal. Appl.
  • 2018
Several block preconditioners for Krylov subspace methods are described and analyzed and the iterative solution of a class of linear systems with double saddle point structure is considered.