We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worst-case analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interior-point methods the overall computational effort is therefore dominated by the least-squares system that must be solved in each iteration. A type of conjugate-gradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugate-gradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.

We continue our study of high-performance algorithms for solving triangular matrix equations. They appear naturally in different condition estimation problems for matrix equations and various eigenspace computations, and as reduced systems in standard algorithms. Building on our successful recursive approach applied to one-sided matrix equations (Part I), we now present novel recursive blocked algorithms for two-sided matrix equations, which include matrix product terms such as AX BT. Examples are the discrete-time standard and generalized Sylvester and Lyapunov equations. The means for achieving high performance is the recursive variable blocking, which has the potential of matching the memory hierarchies of today’s high-performance computing systems, and level-3 computations which mainly are performed as GEMM operations. Different implementation issues are discussed, including the design of efficient new algorithms for two-sided matrix products. We present uniprocessor and SMP parallel performance results of recursive blocked algorithms and routines in the state-of-the-art SLICOT library. Although our recursive algorithms with optimized kernels for the two-sided matrix equations perform more operations, the performance improvements are remarkable, including 10-fold speedups or more, compared to standard algorithms.

We present a Newton--like method for solving algebraic Riccati equations that uses exact line search to improve the sometimes erratic convergence behavior of Newton's method. It avoids the problem of a disastrously large first step and accelerates convergence when Newton steps are too small or too long. The additional work to perform the line search is small relative to the work needed to calculate the Newton step. 1 Introduction We study the generalized continuous--time algebraic Riccati equation (CARE) 0 = R(X) = C T QC +A T XE + E T XA \Gamma (B T XE + S T C) T R \Gamma1 (B T XE + S T C) (1) Here A; E; X 2 IR n\Thetan , B 2 IR n\Thetam , R = R T 2 IR m\Thetam , Q = Q T 2 IR p\Thetap , C 2 IR p\Thetan , and S 2 IR p\Thetam . We will assume that E is nonsingular, Q \Gamma SR \Gamma1 S T 0, and R ? 0 where M ? 0 (M 0) denotes positive (semi-) definite matrices M . In principle, by inverting E, (1) may be reduced to the case E = I . This is conve...

Given a rational matrix G with complex coe#cients and a domain # in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over #, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the poles of G. We apply this result to obtain the parametrized class of all coprime factorizations over # with denominators of minimal McMillan degree n b ---the number of poles of G outside #. Specific choices of the parameters and of # allow us to determine coprime factorizations, as for instance, with polynomial, proper, or stable factors. Further, we consider the case in which the denominator has a certain symmetry, namely it is J all-pass with respect either to the imaginary axis or to the unit circle. We give necessary and su#cient solvability conditions for the problem of coprime factorization with J all-pass denominator of McMillan degree n b and, when a solution exists, we give a construction of the class of coprime factors. When no such solution exists, we discuss the existence of, and give solutions to, coprime factorizations with J all-pass denominators of minimal McMillan degree (>n b ). All the developments are carried out in terms of descriptor realizations associated with rational matrices, leading to explicit and computationally e#cient formulas.

with exact line searches

The determination of the sources of electric activity inside the brain from electric and magnetic measurements on the surface of the head is known to be an ill-posed problem.

We present a kernel-based approach to the classification of a time series of gene expression profiles. Our method takes into account the dynamic evolution over time as well as the temporal characteristics of the data. More specifically, we model the evolution of the gene expression profiles as a Linear Time Invariant (LTI) dynamical system and estimate its model parameters. A kernel on dynamical systems is then used to classify these time series. We successfully test our approach on a published dataset to predict response to drug therapy in Multiple Sclerosis patients. For pharmacogenomics, our method offers a huge potential for advanced computational tools in disease diagnosis, and disease and drug therapy outcome prognosis. 1.

The sensitivity of Sylvester matrix equations relative to perturbations in the coefficientmatricesisstudied. New local perturbations bounds are obtained. Keywords. Perturbation analysis, Sylvester equations. 1 Introduction In this paper we study the sensitivityofSylvester matrix equations (SME) arising in linear systems theory. A new local perturbation bound for SME is obtained, whichisa non-linear, first order homogeneous and tighter than the local bounds based on condition numbers [1] - [5]. The following notations are used later on: R m\Thetan -- the space of real m \Theta n matrices# I n -- the unit n \Theta n matrix # A ? =[a ji ] -- the transpose of the matrix A =[a ij ]# vec(A) 2R mn -- the column-wise vector representation of the matrix A 2R m\Thetan # A\Omega B =[a ij B] -- the Kronecker product of the matrices A and B# k\Deltak 2 -- the spectral (or 2-) norm in R m\Thetan # k:k F -- the Frobenius (or F-) norm in R m\Thetan . The notation `:=' stands for `equal ...