We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The n... more 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
. We present a convergence analysis for a class of inexact infeasible-interior-point methodsfor s... more . We present a convergence analysis for a class of inexact infeasible-interior-point methodsfor solving linear programs. The main feature of inexact methods is that the linear systems definingthe search direction at each interior-point iteration need not be solved to high accuracy. Moreprecisely, we allow that these linear systems are only solved to a moderate relative accuracy in theresidual , but
Abstract The conjugate gradient (CG) algorithm for the solution of Hermitian positive de nite lin... more Abstract The conjugate gradient (CG) algorithm for the solution of Hermitian positive de nite linear systems requires the computation of inner products. On modern architectures, inner products often represent a bottleneck, and it is desirable to reduce or even eliminate all ...
We present an interior-point method for a class of fractional programs with convex constraints. T... more We present an interior-point method for a class of fractional programs with convex constraints. The proposed algorithm converges at a polynomial rate, similarly as in the case of a convex problem, even though fractional programs are only pseudo-convex. Here, the rate of convergence is measured in terms of the area of two-dimensional convex setsCk containing the origin and certain projections
This paper describes the use of partial Pad e approximation to generate stable and passive reduce... more This paper describes the use of partial Pad e approximation to generate stable and passive reduced-order models of linear circuits. For similarly-sized models, partial Pad ebased reduced-oreder modeling has superior moment-matching capabilities than competing techniques based on the Arnoldi process. The paper introduces PVL , an algorithm for computing partial Pad e-based reduced-order models via the Lanczos process. The e ectiveness of this modeling methodology is illustrated by numerical examples.
Abstract: We consider the solution of nonlinear programs with nonlinear semidefinitenessconstrain... more Abstract: We consider the solution of nonlinear programs with nonlinear semidefinitenessconstraints. The need for an efficient exploitation of the cone of positive semidefinite matricesmakes the solution of such nonlinear semidefinite programs more complicated than the solutionof standard nonlinear programs. In particular, a suitable symmetrization procedure needs tobe chosen. The choice of the symmetrization procedure can be shifted in a very
We consider the computation of x := exp(?A)b, where b is a vector and A is a matrix of order N, p... more We consider the computation of x := exp(?A)b, where b is a vector and A is a matrix of order N, possibly nonsymmetric, by means of iterative methods. The algorithm is based on a transpose-free quasi-minimal residual algorithm to e ciently compute the solution of systems (A?z j I)x j = b for several distinct values of the shift z j , and on the partial-fraction representation of rational approximations of the matrix exponential. We outline mappings of the algorithm on a parallel architecture, present numerical experiments, and discuss the computational performance of the algorithm.
Http Dx Doi Org 10 1080 10556780410001654232, May 13, 2010
The well-known positive real lemma characterizes positive realness of transfer functions of timei... more The well-known positive real lemma characterizes positive realness of transfer functions of timeinvariantlinear systems via the solvability of certain linear matrix inequalities. In this paper,we propose an extension of the positive real lemma and the underlying linear matrix inequalitiesto descriptor systems. We show that the solvability of these linear matrix inequalities issufficient and, under an additional assumption, also necessary for
Siam Journal on Scientific and Statistical Computing, Jul 13, 2006
Conjugate gradient-type methods for the solution of large sparse linear systems $Ax = b$ with com... more Conjugate gradient-type methods for the solution of large sparse linear systems $Ax = b$ with complex symmetric coefficient matrices $A = A^T $ are considered. Such linear systems arise in important applications, such as the numerical ...
ABSTRACT Given a self-concordant barrier function for a convex set S, we determine a self-concord... more ABSTRACT Given a self-concordant barrier function for a convex set S, we determine a self-concordant barrier function for the conic hull ~ S of S. As our main result, we derive an "optimal" barrier for ~ S based on the barrier function for S. Important applications of this result include the conic reformulation of a convex problem, and the solution of fractional programs by interior-point methods. The problem of minimizing a convex-concave fraction over some convex set can be solved by applying an interiorpoint method directly to the original nonconvex problem, or by applying an interior-point method to an equivalent convex reformulation of the original problem. Our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases. Key words : Convex set; Conic hull; Barrier function; Interior-point method; Fractional programs; Self-concordance; Convergence rate 1. Conic hulls and fractional programming Let S ` IR n be a given convex s...
We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The n... more 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
. We present a convergence analysis for a class of inexact infeasible-interior-point methodsfor s... more . We present a convergence analysis for a class of inexact infeasible-interior-point methodsfor solving linear programs. The main feature of inexact methods is that the linear systems definingthe search direction at each interior-point iteration need not be solved to high accuracy. Moreprecisely, we allow that these linear systems are only solved to a moderate relative accuracy in theresidual , but
Abstract The conjugate gradient (CG) algorithm for the solution of Hermitian positive de nite lin... more Abstract The conjugate gradient (CG) algorithm for the solution of Hermitian positive de nite linear systems requires the computation of inner products. On modern architectures, inner products often represent a bottleneck, and it is desirable to reduce or even eliminate all ...
We present an interior-point method for a class of fractional programs with convex constraints. T... more We present an interior-point method for a class of fractional programs with convex constraints. The proposed algorithm converges at a polynomial rate, similarly as in the case of a convex problem, even though fractional programs are only pseudo-convex. Here, the rate of convergence is measured in terms of the area of two-dimensional convex setsCk containing the origin and certain projections
This paper describes the use of partial Pad e approximation to generate stable and passive reduce... more This paper describes the use of partial Pad e approximation to generate stable and passive reduced-order models of linear circuits. For similarly-sized models, partial Pad ebased reduced-oreder modeling has superior moment-matching capabilities than competing techniques based on the Arnoldi process. The paper introduces PVL , an algorithm for computing partial Pad e-based reduced-order models via the Lanczos process. The e ectiveness of this modeling methodology is illustrated by numerical examples.
Abstract: We consider the solution of nonlinear programs with nonlinear semidefinitenessconstrain... more Abstract: We consider the solution of nonlinear programs with nonlinear semidefinitenessconstraints. The need for an efficient exploitation of the cone of positive semidefinite matricesmakes the solution of such nonlinear semidefinite programs more complicated than the solutionof standard nonlinear programs. In particular, a suitable symmetrization procedure needs tobe chosen. The choice of the symmetrization procedure can be shifted in a very
We consider the computation of x := exp(?A)b, where b is a vector and A is a matrix of order N, p... more We consider the computation of x := exp(?A)b, where b is a vector and A is a matrix of order N, possibly nonsymmetric, by means of iterative methods. The algorithm is based on a transpose-free quasi-minimal residual algorithm to e ciently compute the solution of systems (A?z j I)x j = b for several distinct values of the shift z j , and on the partial-fraction representation of rational approximations of the matrix exponential. We outline mappings of the algorithm on a parallel architecture, present numerical experiments, and discuss the computational performance of the algorithm.
Http Dx Doi Org 10 1080 10556780410001654232, May 13, 2010
The well-known positive real lemma characterizes positive realness of transfer functions of timei... more The well-known positive real lemma characterizes positive realness of transfer functions of timeinvariantlinear systems via the solvability of certain linear matrix inequalities. In this paper,we propose an extension of the positive real lemma and the underlying linear matrix inequalitiesto descriptor systems. We show that the solvability of these linear matrix inequalities issufficient and, under an additional assumption, also necessary for
Siam Journal on Scientific and Statistical Computing, Jul 13, 2006
Conjugate gradient-type methods for the solution of large sparse linear systems $Ax = b$ with com... more Conjugate gradient-type methods for the solution of large sparse linear systems $Ax = b$ with complex symmetric coefficient matrices $A = A^T $ are considered. Such linear systems arise in important applications, such as the numerical ...
ABSTRACT Given a self-concordant barrier function for a convex set S, we determine a self-concord... more ABSTRACT Given a self-concordant barrier function for a convex set S, we determine a self-concordant barrier function for the conic hull ~ S of S. As our main result, we derive an "optimal" barrier for ~ S based on the barrier function for S. Important applications of this result include the conic reformulation of a convex problem, and the solution of fractional programs by interior-point methods. The problem of minimizing a convex-concave fraction over some convex set can be solved by applying an interiorpoint method directly to the original nonconvex problem, or by applying an interior-point method to an equivalent convex reformulation of the original problem. Our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases. Key words : Convex set; Conic hull; Barrier function; Interior-point method; Fractional programs; Self-concordance; Convergence rate 1. Conic hulls and fractional programming Let S ` IR n be a given convex s...
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Papers by Roland Freund