a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

algorithms for a multi level network optimization problem in this paper we work on a multi level network optimization problem that integrates into the same model important aspects of i discrete facility location ii topological network design and iii network dimensioning potential applications for the model are discussed stressing its growing importance the multi level network optimization problem treated is defined and a mathematical programming formulation is presented we make use of a branch and bound algorithm based on lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem we conduct a set of computational experiments that indicate the quality of the proposed approach

phaselocking in a reaction diffusion system with a linear frequency gradient a simple spatially inhomogeneous oscillatory reaction diffusion equation is studied existence of phaselocked waves is proven with a shooting argument stability is determined numerically applications to intestinal motility and a chemical experiment are described

computational complexity of a cost allocation approach to a fixed cost spanning forest problem we present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem referred to as the fixed cost spanning forest fcsf problem the customers in the fcsf problem represented by nodes in a graph g are in need of service that can be produced at some facilities yet to be constructed the cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers we formulate this cost allocation problem as a cooperative game referred to as the fcsf game in general the core of a fcsf game may be empty however for the case when g is a tree it is shown that the core is not empty moreover we prove that in this case core points can be generated in strongly polynomial time we further provide a nonredundant characterization of the core of the fcsf game defined over a tree in the special case when all nodes are communities this is shown to lead in some instances to a strongly polynomial algorithm for computing the nucleolus

solution of a nonsymmetric algebraic riccati equation from a two dimensional transport model abstract for the steady state solution of an integral differential equation from a two dimensional model in transport theory we shall derive and study a nonsymmetric algebraic riccati equation b xf f x xb x 0 where f i s pd b b i s p d and b b i s pd with a nonnegative matrix p positive diagonal matrices d and nonnegative parameters f b b 1 f and s s 1 f we prove the existence of the minimal nonnegative solution x under the physically reasonable assumption f b s p d d 1 and study its numerical computation by fixed point iteration newton s method and doubling we shall also study several special cases e g when b 0 and p is low ranked then x s 2 uv is low ranked and can be computed using more efficient iterative processes in u and v numerical examples will be given to illustrate our theoretical results

computational experience with a bundle approach for semidefinite cutting plane relaxations of max cut and equipartition we propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities our approach is based on lagrangian duality where the inequalities are dualized and only a basic set of constraints is maintained explicitly this leads to function evaluations requiring to solve a relatively simple semidefinite program our approach provides accurate solutions to semidefinite relaxations of the max cut and the equipartition problem which are not achievable by direct approaches based only on interior point methods

brief paper adaptive optimal control for continuous time linear systems based on policy iteration in this paper we propose a new scheme based on adaptive critics for finding online the state feedback infinite horizon optimal control solution of linear continuous time systems using only partial knowledge regarding the system dynamics in other words the algorithm solves online an algebraic riccati equation without knowing the internal dynamics model of the system being based on a policy iteration technique the algorithm alternates between the policy evaluation and policy update steps until an update of the control policy will no longer improve the system performance the result is a direct adaptive control algorithm which converges to the optimal control solution without using an explicit a priori obtained model of the system internal dynamics the effectiveness of the algorithm is shown while finding the optimal load frequency controller for a power system

lp extreme points and cuts for the fixed charge network design problem many design decisions in transporation communication and manufacturing planning can be modeled as problems of routing multiple commodities between various origin and destination nodes of a directed network each arc of the network is uncapacitated and carries a fixed charge as well as a cost per unit of flow we refer to the general problem of selecting a subset of arcs and routing the required multi commodity flows along the chosen arcs at a minimum total cost as the fixed charge network design problem this paper focuses on strenghthening the linear programming relaxation of a path flow formulation for this problem the considerable success achieved by researchers in solving many related design problems with algorithms that use strong linear programming based lower bounds motivates this study we first develop a convenient characterization of fractional extreme points for the network design linear programming relaxation an auxiliary graph introduced for this characterization also serves to generate two families of cuts that exclude some fractional solutions without eliminating any feasible integer solutions we discuss a separation procedure for one class of inequalities and demonstrate that many of our results generalize known properties of the plant location problem

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

the volume algorithm revisited relation with bundle methods we revise the volume algorithm va for linear programming and relate it to bundle methods when first introduced va was presented as a subgradient like method for solving the original problem in its dual form in a way similar to the serious null steps philosophy of bundle methods va produces green yellow or red steps in order to give convergence results we introduce in va a precise measure for the improvement needed to declare a green or serious step this addition yields a revised formulation rva that is halfway between va and a specific bundle method that we call bva we analyze the convergence properties of both rva and bva finally we compare the performance of the modified algorithms versus va on a set of rectilinear steiner problems of various sizes and increasing complexity derived from real world vlsi design instances

convergence analysis of deflected conditional approximate subgradient methods subgradient methods for nondifferentiable optimization benefit from deflection i e defining the search direction as a combination of the previous direction and the current subgradient in the constrained case they also benefit from projection of the search direction onto the feasible set prior to computing the steplength that is from the use of conditional subgradient techniques however combining the two techniques is not straightforward especially if an inexact oracle is available which can only compute approximate function values and subgradients we present a convergence analysis of several different variants both conceptual and implementable of approximate conditional deflected subgradient methods our analysis extends the available results in the literature by using the main stepsize rules presented so far while allowing deflection in a more flexible way furthermore to allow for diminishing square summable rules where the stepsize is tightly controlled a priori we propose a new class of deflection restricted approaches where it is the deflection parameter rather than the stepsize which is dynamically adjusted using the target value of the optimization sequence for both polyak type and diminishing square summable stepsizes we propose a correction of the standard formula which shows that in the inexact case knowledge about the error computed by the oracle which is available in several practical applications can be exploited in order to strengthen the convergence properties of the method the analysis allows for several variants of the algorithm at least one of them is likely to show numerical performances similar to these of heavy ball subgradient methods popular within backpropagation approaches to train neural networks while possessing stronger convergence properties

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

the linear quadratic optimal regulator for descriptor systems discrete time case abstract in this paper the linear quadratic optimal regulator problem for the discrete time descriptor system ex k 1 ax k bu k where e is in general a singular matrix is investigated the continuous time version of this problem was treated elsewhere the discrete time theory is largely parallel to the continuous time theory but there are several important differences one is that in the discrete time case a fairly general problem can be solved with singular cost matrices the key to the solution as for the continuous time problem is a coordinate system which can be computed by performing a singular value decomposition of e this is called an svd coordinate system the optimization problem is solved by applying hamiltonian minimization to compute the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors in the general singular cost case this trajectory is computed by recursively solving a discrete time riccati equation from the optimal trajectory the optimal descriptor variable feedback gain relating the control to the descriptor variable can be computed several riccati difference equations are derived all of which have the same solution this solution can be used to compute the optimal trajectory and cost for non singular problems even in the finite horizon case the optimal feedback gain can be computed non recursively by solving an eigenvalue eigenvector problem formulated from the system parameters in the svd coordinate system the optimal gain for the infinite horizon steady state problem on the other hand can be computed from the solution of an eigenvector problem involving the untransformed system parameters

primal convergence from dual subgradient methods for convex optimization when solving a convex optimization problem through a lagrangian dual reformulation subgradient optimization methods are favorably utilized since they often find near optimal dual solutions quickly however an optimal primal solution is generally not obtained directly through such a subgradient approach unless the lagrangian dual function is differentiable at an optimal solution we construct a sequence of convex combinations of primal subproblem solutions a so called ergodic sequence which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen we generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions in contrast to previously proposed rules they exploit more information from later subproblem solutions than from earlier ones we evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

dynamic bundle methods lagrangian relaxation is a popular technique to solve difficult optimization problems however the applicability of this technique depends on having a relatively low number of hard constraints to dualize when there are many hard constraints it may be preferable to relax them dynamically according to some rule depending on which multipliers are active from the dual point of view this approach yields multipliers with varying dimensions and a dual objective function that changes along iterations we discuss how to apply a bundle methodology to solve this kind of dual problems our framework covers many separation procedures to generate inequalities that can be found in the literature including but not limited to the most violated inequality we analyze the resulting dynamic bundle method giving a positive answer for its primal dual convergence properties and under suitable conditions show finite termination for polyhedral problems

a factoring approach for the steiner tree problem in undirected networks in communication networks many applications such as video on demand and video conferencing must establish a communications tree that spans a subset k in a vertex set the source node can then send identical data to all nodes in set k along this tree this kind of communication is known as multicast communication a network optimization problem called the steiner tree problem stp is presented to find a least cost multicasting tree in this paper an o e algorithm is presented to find a minimum steiner tree for series parallel graphs where e is the number of edges based on this algorithm we proposed an o 2 2 c e algorithm to solve the steiner tree problem for general graphs where c is the number of applied factoring procedures the c value is strongly related to the topology of a given graph this is quite different from other algorithms with exponential time complexities in k

comparison of bundle and classical column generation when a column generation approach is applied to decomposable mixed integer programming problems it is standard to formulate and solve the master problem as a linear program seen in the dual space this results in the algorithm known in the nonlinear programming community as the cutting plane algorithm of kelley and cheney goldstein however more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard one of them is the bundle method our aim is to illustrate its differences with kelley s method in the process we review alternative stabilization techniques used in column generation comparing them from both primal and dual points of view numerical comparisons are presented for five applications cutting stock which includes bin packing vertex coloring capacitated vehicle routing multi item lot sizing and traveling salesman we also give a sketchy comparison with the volume algorithm

thompson metric method for solving a class of nonlinear matrix equation based on the elegant properties of the thompson metric we prove that the general nonlinear matrix equation x q a f x a q q 1 always has a unique positive definite solution an iterative method is proposed to compute the unique positive definite solution we show that the iterative method is more effective as q increases a perturbation bound for the unique positive definite solution is derived in the end

newton s method for concave operators with resolvent positive derivatives in ordered banach spaces abstract we prove a non local convergence result for newton s method applied to a class of nonlinear equations in ordered real banach spaces the key tools in our approach are special notions of concavity and the spectral theory of resolvent positive operators

quadratic realizability of palindromic matrix polynomials let cl cl 1 cl 2 be a list consisting of a sublist cl 1 of powers of irreducible monic scalar polynomials over an algebraically closed field ff and a sublist cl 2 of nonnegative integers for an arbitrary such list cl we give easily verifiable necessary and sufficient conditions for cl to be the list of elementary divisors and minimal indices of some t palindromic quadratic matrix polynomial with entries in the field ff for cl satisfying these conditions we show how to explicitly construct a t palindromic quadratic matrix polynomial having cl as its structural data that is we provide a t palindromic quadratic realization of cl our construction of t palindromic realizations is accomplished by taking a direct sum of low bandwidth t palindromic blocks closely resembling the kronecker canonical form of matrix pencils an immediate consequence of our in depth study of the structure of t palindromic quadratic polynomials is that all even grade t palindromic matrix polynomials have a t palindromic strong quadratification finally using a particular m o bius transformation we show how all of our results can be easily extended to quadratic matrix polynomials with t even structure

balancing sparse matrices for computing eigenvalues abstract applying a permuted diagonal similarity transform dpap t d 1 to a matrix a before calculating its eigenvalues can improve the speed and accuracy with which the eigenvalues are computed this is often called balancing this paper describes several balancing algorithms for sparse matrices and compares them against each other and the traditional dense algorithm we first discuss our sparse implementation of the dense algorithm our code is faster than the dense algorithm when the density of the matrix is no more than approximately 5 and is much faster for large sparse matrices we next describe a set of randomized balancing algorithms for matrices that are not given explicitly i e given a vector x we can compute only ax and perhaps a t x we motivate these krylov based algorithms using perron frobenius theory results are given comparing the krylov based algorithms to each other and to the sparse and dense direct balancing algorithms looking at norm reduction running times and the accuracy of eigenvalues computed after a matrix is balanced we conclude that sparse balancing algorithms are efficient preconditioners for eigensolvers

a class of iterative methods for solving nonsymmetric algebraic riccati equations arising in transport theory in this paper we consider the nonsymmetric algebraic riccati equation arising in transport theory an important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation we propose a class of iterative methods to solve the vector equation the convergence analysis shows that the sequence of vectors generated by iterative methods with two kinds of specific iterative matrices is monotonically increasing and converges to the minimal positive solution of the vector equation numerical experiments show that the new methods outperform the modified simple iterative method and newton s method

a step toward a unified treatment of continuous and discrete time control problems in this paper we introduce a new approach for a unified theory for continuous and discrete time optimal control problems based on the generalized cayley transformation we also relate the associated discrete and continuous generalized algebraic riccati equations we demonstrate the potential of this new approach by proving a new result for discrete algebraic riccati equations but we also discuss where this new approach as well as all other approaches still is nonsatisfactory we explain a discrepancy observed between the discrete and continuous case and show that this discrepancy is partly due to the consideration of the wrong analogues we also present an idea for an implication scheme that relates general theorems for discrete and continuous control problems

a partition based relaxation for steiner trees the steiner tree problem is a classical np hard optimization problem with a wide range of practical applications in an instance of this problem we are given an undirected graph g v e a set of terminals r subseteq v and non negative costs c e for all edges e in e any tree that contains all terminals is called a steiner tree the goal is to find a minimum cost steiner tree the vertices v backslash r are called steiner vertices the best approximation algorithm known for the steiner tree problem is a greedy algorithm due to robins and zelikovsky siam j discrete math 19 1 122 134 2005 it achieves a performance guarantee of 1 frac ln 3 2 approx 1 55 the best known linear programming lp based algorithm on the other hand is due to goemans and bertsimas math program 60 145 166 1993 and achieves an approximation ratio of 2 2 r in this paper we establish a link between greedy and lp based approaches by showing that robins and zelikovsky s algorithm can be viewed as an iterated primal dual algorithm with respect to a novel lp relaxation the lp used in the first iteration is stronger than the well known bidirected cut relaxation an instance is b quasi bipartite if each connected component of g backslash r has at most b vertices we show that robins and zelikovsky s algorithm has an approximation ratio better than 1 frac ln 3 2 for such instances and we prove that the integrality gap of our lp is between frac 8 7 and frac 2b 1 b 1

the palindromic generalized eigenvalue problem a x λax numerical solution and applications abstract in this paper we propose the palindromic doubling algorithm pda for the palindromic generalized eigenvalue problem pgep a x ax we establish a complete convergence theory of the pda for pgeps without unimodular eigenvalues or with unimodular eigenvalues of partial multiplicities two one or two for eigenvalue 1 some important applications from the vibration analysis and the optimal control for singular descriptor linear systems will be presented to illustrate the feasibility and efficiency of the pda

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

vibration of fast trains palindromic eigenvalue problems and structure preserving doubling algorithms the vibration of fast trains is governed by a quadratic palindromic eigenvalue problem l 2a 1 email protected 0 a 1 x 0 where a 0 a email protected c n x n and a 0 t a 0 accurate and efficient solution can only be obtained using algorithms which preserve the structure of the eigenvalue problem this paper reports on the successful application of the structure preserving doubling algorithms

a shamanskii like acceleration scheme for nonlinear equations at singular roots on applique une variante de la methode de shamanskii pour realiser une methode a convergence superlineaire de resolution d une classe d equations non lineaires ayant une derivee de frechet singuliere a la racine

instabilities of front patterns in reaction diffusion systems abstract recent experiments in chemical reaction diffusion systems with externally imposed concentration gradients may provide access to a host of spatio temporal pattern formation phenomena these systems tend to form steep reaction fronts in response to the external gradient we use singular perturbation techniques normal form calculations and numerical simulations to investigate the existence and the stability of such sustained fronts in one dimensional systems the theoretical predictions are found in quantitative agreement with direct simulations of the hopf bifurcation from steady to periodically oscillating front structures observed in the couette flow reactor also conditions are found under which oscillations of the spatial structure become chaotic in two dimensional systems we address the issue of realizing an experimental situation hitherto unattained a one dimensional chain of coupled oscillators at the onset of the hopf destabilization of the front structure we point out the intimate relationship between the frequency of oscillation of the homogeneous front pattern and the characteristic wavelength of the turing pattern that can develop along the front 2 d 1 2 we comment on subsequent bifurcations that may result from the nonlinear interaction between hopf and turing instabilities as the precursors to spatio temporal chaos in conclusion we emphasize the possibility of probing the transition to mediated defect turbulence in thin film gel reactors

bifurcation to spatially induced chaos in a reaction diffusion system a one dimensional reaction diffusion equation is used to model a class of experimental open chemical systems in which spatiotemporal patterns can be sustained indefinitely numerical simulations for the model in parameter regimes corresponding to experiment reveal only low dimensional behavior the observed bifurcation sequence leads from steady state concentration profiles to temporally chaotic patterns the physical mechanism that causes this behavior is deduced from analysis of the local dynamics localized oscillators appear at each end of the reactor and one of the oscillators acts as a periodic forcing for the other the numerical results are in good qualitative agreement with the experiments

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

a new acceleration method for newton s method at singular points when newton s method is used to find a root of a map from a banach space into itself and the derivative is singular at that root convergence of the newton sequence is in general linear in this paper we give a modification of newton s method that under certain conditions converges superlinearly our method is applicable under more general conditions than other techniques in particular it may be used for certain quadratic problems and problems in which the dimension of the null space of the derivative is larger than one

convergence acceleration for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root linear convergence of the newton sequence to the root is the best

a generalized negative imaginary lemma and riccati based static state feedback negative imaginary synthesis abstract in this paper we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition then an algebraic riccati equation method is given to determine if a system is negative imaginary also a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property the controller synthesis procedure is based on the proposed negative imaginary lemma using this procedure the closed loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty such as in the case of unmodeled spill over dynamics in a lightly damped flexible structure a numerical example is presented to illustrate the usefulness of the results

detecting a definite hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem and associated nearness problems an important class of generalized eigenvalue problems ax lambdabx is those in which a and b are hermitian and some real linear combination of them is definite for the quadratic eigenvalue problem qep lambda 2 a lambdab c x 0 with hermitian a b and c and positive definite a particular interest focuses on problems in which x bx 2 4 x ax x cx is one signed for all non zero x for the positive sign these problems are called hyperbolic and for the negative sign elliptic the important class of overdamped problems arising in mechanics is a sub class of the hyperbolic problems for each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class for definite pairs a b the distance is the crawford number and we derive bisection and level set algorithms both for testing its positivity and for computing it testing hyperbolicity of a qep is shown to reduce to testing a related pair for definiteness the distance to the nearest non hyperbolic or non elliptic n x n qep is shown to be the solution of a global minimization problem with n 1 dearees of freedom numerical results are given to illustrate the theory and algorithms c 2002 published by elsevier science inc

existence and uniqueness of contractive solutions of some riccati equations abstract in this paper we establish criteria for the existence and uniqueness of contractive solutions k of the riccati equation kbk ka dk c 0 under the assumption that the spectra of a and d are disjoint

an efficient schur method to solve the stabilizing problem an efficient solution to the stabilizing problem for both continuous and discrete constant linear systems is presented the real schur form reduction of the system state matrix is used

a framework for analyzing nonlinear eigenproblems and parametrized linear systems associated with an n times n matrix polynomial of degree ell p lambda sum j 0 ell lambda j a j are the eigenvalue problem p lambda x 0 and the linear system problem p omega x b where in the latter case x is to be computed for many values of the parameter omega both problems can be solved by conversion to an equivalent problem l lambda z 0 or l omega z c that is linear in the parameter lambda or omega this linearization process has received much attention in recent years for the eigenvalue problem but it is less well understood for the linear system problem we develop a framework in which more general versions of both problems can be analyzed based on one sided factorizations connecting a general nonlinear matrix function n lambda to a simpler function m lambda typically a polynomial of degree 1 or 2 our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right hand side c and recover the solution x from z for the eigenvalue problem this framework includes many special cases studied in the literature including the vector spaces of pencils mathbb l 1 p and mathbb l 2 p recently introduced by mackey mackey mehl and mehrmann and a class of rational problems we use the framework to investigate the conditioning and stability of the parametrized linear system p omega x b and thereby study the effect of scaling both of the original polynomial and of the pencil l our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice

structured doubling algorithms for weakly stabilizing hermitian solutions of algebraic riccati equations in this paper we propose structured doubling algorithms for the computation of the weakly stabilizing hermitian solutions of the continuous and discrete time algebraic riccati equations respectively assume that the partial multiplicities of purely imaginary and unimodular eigenvalues if any of the associated hamiltonian and symplectic pencil respectively are all even and the c dare and the dual c dare have weakly stabilizing hermitian solutions with property p under these assumptions we prove that if these structured doubling algorithms do not break down then they converge to the desired hermitian solutions globally and linearly numerical experiments show that the structured doubling algorithms perform efficiently and reliably

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

piecewise quadratic approximations in convex numerical optimization we present a bundle method for convex nondifferentiable minimization where the model is a piecewise quadratic convex approximation of the objective function unlike standard bundle approaches the model only needs to support the objective function from below at a properly chosen small subset of points as opposed to everywhere we provide the convergence analysis for the algorithm with a general form of master problem which combines features of trust region stabilization and proximal stabilization taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information numerical results are also reported

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

homoclinic chaos in chemical systems abstract we first focus on complex dynamical phenomena generated by chemical kinetics in homogeneous media we comment on the alternating sequences of periodic and chaotic states observed in some experiments on the belousov zhabotinsky reaction when conducted in a continuously stirred tank reactor we present numerical results obtained with a seven variable oregonator model which reproduce most of the features of the experimental sequences we discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity we construct poincare map models which reduce to either multi humped or multi branched 1d maps in the strong area contraction limit we report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of spiral type strange attractors satistifies the symbolic dynamics predicted by sil nikov theory of homoclinic chaos we then investigate spatio temporal pattern forming phenomena in a one dimensional reaction diffusion system with equal diffusion coefficients when imposing a concentration gradient through the system this model mimics the sustained stationary and periodically oscillating front structures observed in recent experiments conducted in an open couette flow reactor we emphasize the possibility that the oscillations of the spatial structure become chaotic we report on numerical simulations of a diffusion induced intermittent bursting phenomenon that is likely to be observed in bench experiments we elaborate on the interpretation of this intermittent occurrence of spatially localized structures in an extended system in terms of sil nikov s homoclinic chaos

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

transformations between discrete time and continuous time algebraic riccati equations we introduce a transformation between the discrete time and continuous time algebraic riccati equations we show that under mild conditions the two algebraic riccati equations can be transformed from one to another and both algebraic riccati equations share common hermitian solutions the transformation also sets up the relations about the properties commonly in system and control setting that are imposed in parallel to the coefficient matrices and hermitian solutions of two algebraic riccati equations the transformation is simple and all the relations can be easily derived we also introduce a generalized transformation that requires weaker conditions the proposed transformations may provide a unified tool to develop the theories and numerical methods for the algebraic riccati equations and the associated system and control problems

on iterative methods for the quadratic matrix equation with m matrix in this paper we consider newton s method and bernoulli s method for a quadratic matrix equation arising from an overdamped vibrating system by introducing m matrix to this equation we provide a sufficient condition for the existence of the primary solution moreover we show that newton s method and bernoulli s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition

block minimal bases l ifications of matrix polynomials abstract the standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil known as a strong linearization this process transforms the problem into an equivalent generalized eigenvalue problem however there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials this has motivated the idea of a strong l ification of a matrix polynomial which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial we present in this work a novel method for constructing strong l ifications of matrix polynomials of size m n and grade d when l d and l divides nd or md this method is based on a family called strong block minimal bases matrix polynomials and relies heavily on properties of dual minimal bases we show how strong block minimal bases l ifications can be constructed from the coefficients of a given matrix polynomial p we also show that these l ifications satisfy many desirable properties for numerical applications they are strong l ifications regardless of whether p is regular or singular the minimal indices of the l ifications are related to those of p via constant uniform shifts and eigenvectors and minimal bases of p can be recovered from those of any of the strong block minimal bases l ifications in the special case where l divides d we introduce a subfamily of strong block minimal bases matrix polynomials named block kronecker matrix polynomials which is shown to be a fruitful source of companion l ifications

spatiotemporal pattern formation and chaos in passive optical systems abstract the interaction and competition of transverse cavity modes in passive optical systems can lead to the formation of spatial field structures whose symmetry differs from that of the empty cavity while the time behaviour can become irregular after having introduced dynamical equations in which the spatial dependence of the electric field is described by frequency degenerate gauss laguerre gl modes the stationary solutions corresponding to a single or many modes are calculated and their stability is discussed a scan of the parameter space allows for the identification of several interesting dynamical regimes an extensive study of the chaotic states is performed in order to understand the relation between gl mode excitation and the relevant active degrees of freedom based on this analysis the conditions for the onset of spatiotemporal chaos for its characterisation and experimental observation are finally discussed

low rank approximation to the solution of a nonsymmetric algebraic riccati equation from transport theory abstract we consider the solution of the large scale nonsymmetric algebraic riccati equation xcx xd ax b 0 from transport theory juang 1995 with m d c b a r 2 n 2 n being a nonsingular m matrix in addition a d are rank 1 updates of diagonal matrices with the products a 1 u a u d 1 v and d v computable in o n complexity for some vectors u and v and b c are rank 1 the structure preserving doubling algorithm by guo et al 2006 is adapted with the appropriate applications of the sherman morrison woodbury formula and the sparse plus low rank representations of various iterates the resulting large scale doubling algorithm has an o n computational complexity and memory requirement per iteration and converges essentially quadratically as illustrated by the numerical examples

structured backward error for palindromic polynomial eigenvalue problems a detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems ppep left sum ell 0 d a ell lambda ell right x 0 quad a d ell varepsilon a ell star quad rm for ell 0 1 ldots lfloor d 2 rfloor is one of the two actions transpose and conjugate transpose and varepsilon in pm 1 each of them has its application background with the case star taking transpose and e 1 attracting a great deal of attention lately because of its application in the fast train modeling computable formulas and bounds for the structured backward errors are obtained the analysis reveals distinctive features of ppep from general polynomial eigenvalue problems pep investigated by tisseur linear algebra appl 309 339 361 2000 and by liu and wang appl math comput 165 405 417 2005

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

survivable networks linear programming relaxations and the parsimonious property we consider the survivable network design problem the problem of designing at minimum cost a network with edge connectivity requirements as special cases this problem encompasses the steiner tree problem the traveling salesman problem and thek edge connected network design problem we establish a property referred to as the parsimonious property of the linear programming lp relaxation of a classical formulation for the problem the parsimonious property has numerous consequences for example we derive various structural properties of these lp relaxations we present some algorithmic improvements and we perform tight worst case analyses of two heuristics for the survivable network design problem

efficient methods for solving a nonsymmetric algebraic riccati equation arising in stochastic fluid models we consider the nonsymmetric algebraic riccati equation xm12x xm11 m22x m21 0 where m11 m12 m21 m22 are real matrices of sizes n n n m m n m m respectively and m mij 2i j 1 is an irreducible singular m matrix with zero row sums the equation plays an important role in the study of stochastic fluid models where the matrix m is the generator of a markov chain the solution of practical interest is the minimal nonnegative solution this solution may be found by basic fixed point iterations newton s method and the schur method however these methods run into difficulties in certain situations in this paper we provide two efficient methods that are able to find the solution with high accuracy even for these difficult situations

the volume algorithm producing primal solutions with a subgradient method we present an extension to the subgradient algorithm to produce primal as well as dual solutions it can be seen as a fast way to carry out an approximation of dantzig wolfe decomposition this gives a fast method for producing approximations for large scale linear programs it is based on a new theorem in linear programming duality we present successful experience with linear programs coming from set partitioning set covering max cut and plant location

new approaches for optimizing over the semimetric polytope the semimetric polytope is an important polyhedral structure lying at the heart of several hard combinatorial problems therefore linear optimization over the semimetric polytope is crucial for a number of relevant applications building on some recent polyhedral and algorithmic results about a related polyhedron the rooted semimetric polytope we develop and test several approaches based over lagrangian relaxation and application of non differentiable optimization algorithms for linear optimization over the semimetric polytope we show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state of the art general purpose linear programming technology in some cases good estimates of the dual optimal solution but not of the primal solution can be obtained even quicker

palindromic quadratization and structure preserving algorithm for palindromic matrix polynomials of even degree in this paper we propose a palindromic quadratization approach transforming a palindromic matrix polynomial of even degree to a palindromic quadratic pencil based on the mathcal s mathcal s 1 transform and patel s algorithm the structure preserving algorithm can then be applied to solve the corresponding palindromic quadratic eigenvalue problem numerical experiments show that the relative residuals for eigenpairs of palindromic polynomial eigenvalue problems computed by palindromic quadratized eigenvalue problems are better than those via palindromic linearized eigenvalue problems or texttt polyeig in matlab

lagrangian relaxation via ballstep subgradient methods we exhibit useful properties of ballstep subgradient methods for convex optimization using level controls for estimating the optimal value augmented with simple averaging schemes they asymptotically find objective and constraint subgradients involved in optimality conditions when applied to lagrangian relaxation of convex programs they find both primal and dual solutions and have practicable stopping criteria up until now similar results have only been known for proximal bundle methods and for subgradient methods with divergent series stepsizes whose convergence can be slow encouraging numerical results are presented for large scale nonlinear multicommodity network flow problems

relationships between discrete time and continuous time algebraic riccati inequalities abstract the bilinear transformation is used to establish a direct relationship between a discrete time algebraic riccati inequality dari and an associated continuous time algebraic riccati inequality cari it is shown that under mild conditions the dari is solvable if and only if the corresponding cari is solvable the relationship between the dari and the cari is then used to translate the general solvability conditions for a cari given by scherer into analogous conditions for the dari it is shown how such conditions can be applied to determine the solvability of a discrete time h control problem whose solution set is characterized by two daris

lagrangian heuristic for a class of the generalized assignment problems a lagrangian based heuristic is proposed for many to many assignment problems taking into account capacity limits for task and agents a modified lagrangian bound studied earlier by the authors is presented and a greedy heuristic is then applied to get a feasible lagrangian based solution the latter is also used to speed up the subgradient scheme to solve the modified lagrangian dual problem a numerical study is presented to demonstrate the efficiency of the proposed approach

an exact line search method for solving generalized continuous time algebraic riccati equations we present a newton like method for solving algebraic riccati equations that uses an 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

a stabilized structured dantzig wolfe decomposition method we discuss an algorithmic scheme which we call the stabilized structured dantzig wolfe decomposition method for solving large scale structured linear programs it can be applied when the subproblem of the standard dantzig wolfe approach admits an alternative master model amenable to column generation other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron stabilization is achieved by the same techniques developed for the standard dantzig wolfe approach and it is equally useful to improve the performance as shown by computational results obtained on an application to the multicommodity capacitated network design problem

reducing matching to polynomial size linear programming the question of whether the maximum weight matching problem can be reduced to a linear program of polynomial size is studied a partial answer to it is given i e it is shown that the chinese postman problem and optimum matching reduces to a sequence of o m 2 log n minimum mean cycle problems it is shown that this last problem can be formulated as a linear program of polynomial size this gives a polynomial algorithm for matching based on any polynomial method for linear programming a combinatorial algorithm for finding minimum mean cycles in undirected graphs is also given

two well known properties of subgradient optimization the subgradient method is both a heavily employed and widely studied algorithm for non differentiable optimization nevertheless there are some basic properties of subgradient optimization that while well known to specialists seem to be rather poorly known in the larger optimization community this note concerns two such properties both applicable to subgradient optimization using the divergent series steplength rule the first involves convergence of the iterative process and the second deals with the construction of primal estimates when subgradient optimization is applied to maximize the lagrangian dual of a linear program the two topics are related in that convergence of the iterates is required to prove correctness of the primal construction scheme

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

constructing strong l ifications from dual minimal bases abstract we provide an algorithm for constructing strong l ifications of a given matrix polynomial p of degree d and size m n using only the coefficients of the polynomial and the solution of linear systems of equations a strong l ification of p is a matrix polynomial of degree l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial p all explicit constructions of strong l ifications introduced so far in the literature have been limited to the case where l divides d though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible based on recent developments on dual polynomial minimal bases we present a general construction of strong l ifications for wider choices of the degree l namely when l divides one of nd or md and d l in the case where l divides nd respectively md the strong l ifications we construct allow us to easily recover the minimal indices of p in particular we show that they preserve the left resp right minimal indices of p and the right resp left minimal indices of the l ification are the ones of p increased by d l each moreover in the particular case l divides d the new method provides a companion l ification that resembles very much the companion l ifications already known in the literature

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

bundle methods for sum functions with easy components applications to multicommodity network design we propose a version of the bundle scheme for convex nondifferentiable optimization suitable for the case of a sum function where some of the components are easy that is they are lagrangian functions of explicitly known compact convex programs this corresponds to a stabilized partial dantzig wolfe decomposition where suitably modified representations of the easy convex subproblems are inserted in the master problem as an alternative to iteratively inner approximating them by extreme points thus providing the algorithm with exact information about a part of the dual objective function the resulting master problems are potentially larger and less well structured than the standard ones ruling out the available specialized techniques and requiring the use of general purpose solvers for their solution this strongly favors piecewise linear stabilizing terms as opposed to the more usual quadratic ones which in turn may have an adverse effect on the convergence speed of the algorithm so that the overall performance may depend on appropriate tuning of all these aspects yet very good computational results are obtained in at least one relevant application the computation of tight lower bounds for fixed charge multicommodity min cost flow problems

connectivity upgrade models for survivable network design disruptions in infrastructure networks to transport material energy and information can have serious economic and even catastrophic consequences since these networks require enormous investments network service providers emphasize both survivability and cost effectiveness in their topological design decisions this paper addresses the survivable network design problem a core model incorporating the cost and redundancy trade offs facing network planners using a novel connectivity upgrade strategy we develop several families of inequalities to strengthen a multicommodity flow based formulation for the problem and show that some of these inequalities are facet defining by increasing the linear programming lower bound the valid inequalities not only lead to better performance guarantees for heuristic solutions but also accelerate exact and approximate solution methods we also consider a heuristic strategy that sequentially rounds the fractional values starting with the linear programming solution to our strong model extensive computational tests confirm that the valid inequalities added via a cutting plane algorithm and the heuristic procedure are very effective and their performance is robust to changes in the network dimensions and connectivity structure our solution approach generates tight lower and upper bounds with average gaps that are less than 1 2 for various problem sizes and connectivity requirements

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

on equivalence of pencils from discrete time and continuous time control we study two matrix pencils that arise respectively in discrete time and continuous time optimal and robust control we introduce a one to one transformation between these two pencils we show that for the pencils under the transformation their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related the eigen structures of the pencils under consideration have strong connections with the associated control problems our result may be applied to connect the discrete time and continuous time control problems and eventually lead to a unified treatment of these two types of control problems

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs