Eigenvalue Bounds for Double Saddle-Point Systems

Abstract

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 zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.

A Preconditioned Inexact Active-Set Method for Large-Scale Nonlinear Optimal Control Problems

“…In this interesting paper, the authors also demonstrate the effect of approximating the blocks A 1 , S 1 , S 2 within the block-diagonal preconditioner, further highlighting the effectiveness of this strategy. We also highlight [8,Thm. 3.3], which analyzes the case A 2 = 0.…”

“…5.3 match those obtained in Thm. 3.4 of the recent preprint [8], using a different structure of proof. In this interesting paper, the authors also demonstrate the effect of approximating the blocks A 1 , S 1 , S 2 within the block-diagonal preconditioner, further highlighting the effectiveness of this strategy.…”

“…Preconditioning PDE-constrained optimization problems has been a subject of considerable interest of late (see, e.g., [53,50,60,44,42,43,45,40]), however relatively few such methods have made use of the double saddle-point structure. We refer the reader to recent, related work [37,55,41,8]. We provide theoretical results on preconditioners for double saddle-point systems and, having arranged the linear systems obtained from the active-set approach to this form, we present efficient and flexible approximations to be applied within a Krylov-subspace solver.…”