A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system.

A standard approach to model reduction of large-scale higher-order linear dynamical
systems is to rewrite the system as an equivalent first-order system and then employ
Krylov-subspace techniques for model reduction of first-order systems. This paper presents
some results about the structure of the block-Krylov subspaces induced by the matrices of
such equivalent first-order formulations of higher-order systems. Two general classes of
matrices, which exhibit the key structures of the matrices of first-order formulations of
higher-order systems, are introduced. It is proved that for both classes, the block-Krylov
subspaces induced by the matrices in these classes can be viewed as multiple copies of
certain subspaces of the state space of the original higher-order system.

A standard approach to reduced-order modeling of higher-order linear dynamical
systems is to rewrite the system as an equivalent first-order system and then employ
Krylov-subspace techniques for reduced-order modeling of first-order systems. While this
approach results in reduced-order models that are characterized as Pade-type or even true
Pade approximants of the system's transfer function, in general, these models do not
preserve the form of the original higher-order system. In this paper, we present a new
approach to reduced-order modeling of higher-order systems based on projections onto
suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace
techniques to an equivalent first-order system. We show that the resulting reduced-order
models preserve the form of the original higher-order system. While the resulting
reduced-order models are no longer optimal in the Pade sense, we show that they still
satisfy a Pade-type approximation property. We also introduce the notion of Hermitian
higher-order linear dynamical systems, and we establish an enhanced Pade-type approximation
property in the Hermitian case.

We consider the solution of nonlinear programs with nonlinear semidefiniteness
constraints. The need for an efficient exploitation of the cone of positive semidefinite
matrices makes the solution of such nonlinear semidefinite programs more complicated than
the solution of standard nonlinear programs. In particular, a suitable symmetrization
procedure needs to be chosen for the linearization of the complementarity condition. The
choice of the symmetrization procedure can be shifted in a very natural way to certain
linear semidefinite subproblems, and can thus be reduced to a well-studied problem. The
resulting sequential semidefinite programming (SSP) method is a generalization of the
well-known SQP method for standard nonlinear programs. We present a sensitivity result for
nonlinear semidefinite programs, and then based on this result, we give a self-contained
proof of local quadratic convergence of the SSP method. We also describe a class of
nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report
results of some numerical experiments with the SSP method applied to problems in that
class.

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known sequential quadratic programming method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.