Under the entrywise dominance partial ordering, T.L. Markham and R.L. Smith obtained a Schur complement inequality for the Hadamard product of two tridiagonal totally nonnegative matrices. Applying the properties of the Hadamard core of totally nonnegative matrices, the Schur complement inequalities for the Hadamard product of totally nonnegative matrices is obtained, which extends those of T.L...

Let A be a nonnegative idempotent matrix. It is shown that the Schur complement of a submatrix, using the Moore-Penrose inverse, is a nonnegative idempotent matrix if the submatrix has a positive diagonal. Similar results for the Schur complement of any submatrix of A are no longer true in general.

Deterministic sample average approximations of stochastic programming problems with recourse are suitable for a scenario-based parallelization. In this paper the parallelization is obtained by using an interior-point method and a Schur complement mechanism for the interior-point linear systems. However, the direct linear solves involving the dense Schur complement matrix are expensive, and adve...

We describe an active-set, dual-feasible Schur-complement method for quadratic programming (QP) with positive definite Hessians. The formulation of the QP being solved is general and flexible, and is appropriate for many different application areas. Moreover, the specialized structure of the QP is abstracted away behind a fixed KKT matrix called Ko and other problem matrices, which naturally le...

Let A be a nonnegative idempotent matrix. It is shown that the Schur complement of a submatrix, using the Moore-Penrose inverse, is a nonnegative idempotent matrix if the submatrix has a positive diagonal. Similar results for the Schur complement of any submatrix of A are no longer true in general.

This paper introduces a robust preconditioner for general sparse symmetric matrices, that is based on low-rank approximations of the Schur complement in a Domain Decomposition (DD) framework. In this “Schur Low Rank” (SLR) preconditioning approach, the coefficient matrix is first decoupled by DD, and then a low-rank correction is exploited to compute an approximate inverse of the Schur compleme...

AbRtract This paper deals with the domain decomposition-based preconditioned conjugate gradient method. The Schur complement is expressed as a function of & simple interface matrix. This function is approximated by a simple rational function to generate a simple matrix that is then used 8.8 & preconditioner for the Schur complement. Extensive experiments are performed to examine the effectivene...

We present a class of domain decomposition (DD) preconditioners for the solution of elliptic linear-quadratic optimal control problems. Our DD preconditioners are extensions of Neumann–Neumann DD preconditioners, which have been successfully applied to the solution of single PDEs. The DD preconditioners are based on a decomposition of the optimality conditions for the elliptic linear-quadratic ...

The formula (1) was first used by Schur [22], but the idea of the Schur complement goes back to Sylvester (1851), and the term Schur complement was introduced by E. Haynsworth [16]. In the beginning Schur complements were used in the theory of matrices. M.G. Krein [19] and W.N. Anderson and G.E. Trapp [4] extended the notion of Schur complements of matrices to shorted operators in Hilbert space...