kalman decomposition for implicit linear systems the kalman decomposition for a general class of implicit systems is introduced and studied the properties of strong reachability observability acceptance of all inputs and outer sequences output uniqueness uniqueness and trajectory observability of the original system are related to the properties of its subsystems obtained via the kalman decomposition the problems of transfer relation regularizability by feedback output feedback and output injection are studied by means of this decomposition

on singular implicit linear dynamical systems we investigate properties of existence unicity representation of the causal solutions of implicit linear systems or generalized systems when the underlying matrix pencil is singular we relate the geometric and the algebraic approaches the main conclusion is that if the underlying matrix pencil is column singular i e has a nonempty set of column minimal indices the causal solutions when they exist can exactly be represented as the output of a classical two player dynamical system where the second player accounts for the nonunicity properties of the equivalent system are related to those of the singular matrix pencils made with the given matrices

disturbance decoupled estimation for linear differential algebraic systems we study the disturbance decoupled estimation problem for linear differential algebraic systems which are not necessarily regular we introduce the notion of partial state observers following a rec

computing the regularization of a linear differential algebraic system abstract we study the regularization problem for linear differential algebraic systems as an improvement of former results we show that any system can be regularized by a combination of state space and input space transformations behavioral equivalence transformations and a reorganization of variables the additional state feedback which is needed in earlier publications is shown to be superfluous we provide an algorithmic procedure for the construction of the regularization and discuss computational aspects

an optimization algorithm for the vehicle routing problem with time windows based on lagrangian relaxation our paper presents a new optimization method for the vehicle routing problem with time windows vrptw the vrptw is a generalization of the vehicle routing problem where the service of a customer must start within a given time interval a so called time window our method is based on a lagrangian relaxation of the constraint set requiring that each customer must be serviced the master problem consists of finding the optimal lagrangian multipliers and the subproblem is a shortest path problem with time windows and capacity constraints the optimal multipliers are found using a method exploiting the benefits of subgradient methods as well as a bundle method the method has been implemented and tested on a series of well known benchmark problems of size up to 100 customers our algorithm turns out to be very competitive compared to algorithms considered in the literature and we have succeeded in solving several previously unsolved problems

solvability and stability of a power system dae model abstract the dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators the swing equations resulting in a differential algebraic equation dae for general daes solvability is not guaranteed in general in the linear case the coefficient matrices have to satisfy a certain regularity condition we derive a solvability characterization for the linearized power system dae solely in terms of the network topology as an extension to previous result we allow for higher order generator dynamics furthermore we show that any solvable power system dae is automatically of index one which means that it is also numerically well posed finally we show that any solvable power system dae is stable but not asymptotically stable

a unified set of resequencing algorithms a general algorithm for resequencing equations and finite element meshes is presented this algorithm incorporates in a unified form the most successful resequencing schemes namely the sloan gibbs king gk and gibbs poole stockmeyer gps algorithms as well it provides a context within which the development of new algorithms may take place with a minimum of change it is easily programmed and has been shown to speed up the execution of the original gk algorithm

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

an algorithm for profile and wavefront reduction of sparse matrices summary an algorithm for reducing the profile and wavefront of a sparse matrix is described the scheme is applicable to any sparse matrix which has a symmetric pattern of zeros and may be used to generate efficient labellings for finite element grids in particular it is suitable for generating efficient labellings for profile and frontal solution schemes empirical evidence obtained from analysis of the 30 test problems collected by everstine suggests that the new algorithm is superior to existing methods for profile and wavefront reduction it is fast requires only a small amount of memory and is simple to program

the pencil se a and controllability observability for generalized linear systems a geometric approach in the first part of the paper we study the pencil se a by using some geometric concepts given by bernhard siam j control optim 20 1982 pp 612 6331 and some new ones which are introduced here such concepts are then used to obtain an alternative characterization for the finite and infinite zero structure of the pencil and to construct a polynomial basis for ker se a necessary and sufficient conditions for the rows or and the columns of the pencil to be linearly independent over the ring of the polynomials are also given the main geometric properties of a regular pencil are presented including the identification of the subspace in which the impulsive response of the autonomous generalized linear system e dot x ax takes place in the second part we consider the generalized linear system e dot x ax bu y cx and we give necessary and sufficient conditions for the infinite zeros of the regular pencil se a to be controllable and observable

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

discounted cost linear quadratic gaussian control for descriptor systems we consider the linear quadratic gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon based on recent results from the deterministic frame

a hybrid algorithm for reducing matrix bandwidth a hybrid algorithm for reducing the bandwidth of symmetric matrices is described in terms of a finite element grid the new algorithm produces significantly lower bandwidths than either the commonly used gibbs poole stockmeyer gps or cuthill mckee cm algorithms with run times comparable to the original cm algorithm the new hybrid algorithm uses the gps algorithm as a preprocessor to provide a good initial node numbering for the author s node shuffling algorithm which accounts for variable degrees of freedom per node dof node the hybrid algorithm was tested on the 30 benchmark problems that were compiled by everstine and on 10 supplemental problems with variable dof node bandwidths and cpu times are presented for all problems

an arithmetic for matrix pencils theory and new algorithms this paper introduces arithmetic like operations on matrix pencils the pencil arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case these operations give an unusual perspective on a variety of pencil related computations we derive generalizations of monodromy matrices and the matrix exponential a new algorithm for computing a pencil arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces

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

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

an outer approximate subdifferential method for piecewise affine optimization piecewise affine functions arise from lagrangian duals of integer programming problems and optimizing them provides good bounds for use in a branch and bound method methods such as the subgradient method and bundle methods assume only one subgradient is available at each point but in many situations there is more information available we present a new method for optimizing such functions which is related to steepest descent but uses an outer approximation to the subdifferential to avoid some of the numerical problems with the steepest descent approach we provide convergence results for a class of outer approximations and then develop a practical algorithm using such an approximation for the compact dual to the linear programming relaxation of the uncapacitated facility location problem we make a numerical comparison of our outer approximation method with the projection method of conn and cornuejols and the bundle method of schramm and zowe

the gap distance to the set of singular matrix pencils abstract we study matrix pencils s e a using the associated linear subspace ker a e the distance between subspaces is measured in terms of the gap metric in particular we investigate the gap distance of a regular matrix pencil to the set of singular pencils and provide upper and lower bounds for it a relation to the distance to singularity in the frobenius norm is provided

regularization of linear time invariant differential algebraic systems we derive equivalent criteria for the existence of a feedback ensuring that a given linear and time invariant differential algebraic control system is regular or autonomous respectively algebraic and geometric criteria are stated in terms of the involved matrices and the augmented wong sequences for systems which are not regularizable by feedback we show that an additional behavioral equivalence transformation and a reorganization of input and state variables leads to a regular system the index of which is at most one this procedure is known however our approach allows for a detailed characterization of the resulting regular system

the quasi weierstras form for regular matrix pencils regular linear matrix pencils a e k n n where k q r or c and the associated differential algebraic equation dae e u x ax are studied the wong sequences of subspaces are investigate and invoked to decompose the k n into v w where any bases of the linear spaces v and wtransform the matrix pencil into the quasi weierstras form the quasi weierstras form of the matrix pencil decouples the original dae into the underlying ode and the pure dae or in other words decouples the set of initial values into the set of consistent initial values vand pure inconsistent initial values w 0 furthermore vand ware spanned by the general ized eigenvectors at the finite and infinite eigenvalues resp the quasi weierstras form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of a e lead to the well known weierstras form so the latter can be viewed as a generalized jordan form finally it is shown how eigenvector chains constitute a basis for the solution space of e u x ax

kalman type decompostion for implicit linear discrete time systems and its applications a kalman type decomposition for implicit linear discrete time systems is studied in detail it is shown that such a decomposition can be used to study the problem of regularizability

near minimal matrix profiles and wavefronts for testing nodal resequencing algorithms the speeds of algorithms that are specifically designed to solve sparse matrix equations depend on the ordering of the unknowns because it is difficult to know what a good ordering is many resequencing algorithms have been developed to reorder the equations in a manner that minimizes the execution time of the solver being used there is no theoretical way of evaluating resequencing algorithms but four widely used algorithms cuthill mckee gibbs poole stockmeyer levy gibbs king have been compared with one another on the basis of their performance on a set of benchmark test problems this paper reports what we believe to be are minimal or near minimal matrix profiles and wavefronts for the benchmark problems comparisons of the minimal results with those produced by the widely used resequencing algorithms show that they produce profiles typically a few tens of per cent greater than minimal but 50 per cent to 100 per cent greater on two problem types the algorithm that produced the near minimal results used a simulated annealing technique and is far too slow for general use

duality of matrix pencils wong chains and linearizations we consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils one is dual pencils a pair of pencils with the same regular part and related singular structures they were introduced by v kublanovskaya in the 1980s the other is wong chains families of subspaces associated with possibly singular matrix pencils that generalize jordan chains they were introduced by k t wong in the 1970s together dual pencils and wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications especially in the context of linearizations of matrix polynomials we first give a self contained introduction to these two concepts using modern language and extending them to a more general form we describe the relation between them and show how they act on the kronecker form of a pencil and on spectral and singular structures eigenvalues eigenvectors and minimal bases then we present several new applications of these results to more recent topics in matrix pencil theory including constraints on the minimal indices of singular hamiltonian and symplectic pencils new sufficient conditions under which pencils in l1 l2 linearization spaces are strong linearizations a new perspective on fiedler pencils and a link between the moller stetter theorem and some linearizations of matrix polynomials

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

canonical projectors for linear differential algebraic equations abstract this paper aims at improving the decoupling projector chain approach for daes to provide complete decouplings these special canonical projectors are shown to form exactly the spectral projection related to the matrix coefficient pair given by the dae further they prove their value when describing the constraint manifolds of the dae

on differential algebraic equations in infinite dimensions abstract we consider a class of differential algebraic equations daes with index zero in an infinite dimensional hilbert space we define a space of consistent initial values which lead to classical continuously differential solutions for the associated dae moreover we show that for arbitrary initial values we obtain mild solutions for the associated problem we discuss the asymptotic behaviour of solutions for both problems in particular we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil

descriptor systems expanded descriptor equation and markov parameters generalized concepts of solvability and conditionability are presented for descriptor systems and tests are given for these generalized properties an expression is derived for descriptor system markov parameters the notion of expanded descriptor equation is presented

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

kalman controllability decompositions for differential algebraic systems abstract we study linear differential algebraic control systems and investigate decompositions with respect to controllability properties we show that the augmented wong sequences can be exploited for a transformation of the system into a kalman controllability decomposition kcd the kcd decouples the system into a completely controllable part an uncontrollable part given by an ordinary differential equation and an inconsistent part which is behaviorally controllable but contains no completely controllable part this decomposition improves a known kcd from a behavioral point of view we conclude the paper with some features of the kcd in the case of regular systems

disturbance decoupling by behavioral feedback for linear differentialalgebraic systems we study disturbance decoupling for linear differentialalgebraic systems which are not necessarily regular compared to previous approaches where state feedback is used we use the concept of behavioral feedback which allows to study a larger class of systems we derive geometric characterizations for solvability of the disturbance decoupling problem following the classical approach exploiting the freedom in the choice of the behavioral feedback we show that whenever disturbance decoupling can be achieved by behavioral feedback we may additionally achieve autonomous zero dynamics finally we solve lebrets twenty year old open problem concerning disturbance decoupling with output uniqueness using behavioral feedback

on the regularizability of singular systems the property of regularity of singular systems is not a feedback invariant to correct this deficiency a new property of regularizability is defined and geometric tests are given for it in terms of system matrices regularizability is shown to be the natural extension of regularity which is a condition on the homogeneous system to controlled singular systems definitions of controllability and reachability are modified depending on regularizability rather than regularity a brief comparison of proportional and proportional plus derivative feedback laws in the context of making the closed loop system regular regular and reachable and regular and controllable is also given dynamical interpretations of these properties are also presented

a new approach for crew pairing problems by column generation with an application to air transportation we propose a new approach to crew pairing problems arising in the context of airline companies the problem is first formulated as a large scale set covering problem with many colums each column representing a valid crew pairing we then suggest a solution procedure for the continuous relaxation of this large scale problem based on generalized linear programming in which the column generation subproblem is shown to be equivalent to a shortest path problem in an associated graph computational results obtained on a series of real problems involving up to 329 flight segments are reported confirming both computational efficiency and practical applicability of the new approach indeed not only were the resulting solutions observed to be integral for most test problems but average savings of about 4 to 5 over the best available hand built solutions were shown to be obtained

controllability of switched differential algebraic equations abstract we study controllability of switched differential algebraic equations we are able to establish a controllability characterization where we assume that the switching signal is known the characterization takes into account possible jumps induced by the switches it turns out that controllability not only depends on the actual switching sequence but also on the duration between the switching times

distribution systems design with role dependent objectives in this paper we introduce flexible models for capacitated discrete location problems we describe three different points of view of a location problem in a logistics system different mathematical programming formulations are presented illustrated by examples and compared using a battery of test problems extensive computational tests are done showing the potentials and limits of this kind of resolution approach

solutions of differential algebraic equations as outputs of lti systems application to lq control problems abstract in this paper we propose a unified geometric framework for representing all solutions of a linear time invariant differential algebraic equation dae lti as outputs of classical linear time invariant ordinary differential equations ode lti the proposed framework is then used to solve an lq optimal control problem for dae ltis with rectangular matrices

determinability and state estimation for switched differential algebraic equations the problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential algebraic equations daes since this system class imposes time varying dynamic and static in the form of algebraic constraints relations on the evolution of state trajectories an appropriate notion of observability is presented which accommodates these phenomena based on this notion we first derive a formula for the reconstruction of the state of the system where we explicitly obtain an injective mapping from the output to the state in practice such a mapping may be difficult to realize numerically and hence a class of estimators is proposed which ensures that the state estimate converges asymptotically to the real state of the system

detectability and observer design for switched differential algebraic equations this paper studies detectability for switched linear differential algebraic equations daes and its application to the synthesis of observers which generate asymptotically converging state estimates equating detectability to asymptotic stability of zero output constrained state trajectories and building on our work on interval wise observability we propose the notion of interval wise detectability if the output of the system is constrained to be identically zero over an interval then the norm of the corresponding state trajectories scales down by a certain factor at the end of that interval conditions are provided under which the interval wise detectability leads to asymptotic stability of zero output constrained state trajectories an application is demonstrated in designing state estimators decomposing the state into observable and unobservable components we show that if the observable component of the system is reset appropriately and persistently then the estimation error converges to zero asymptotically under the interval wise detectability assumption

the structure of linear relations in euclidean spaces the structure of a linear relation multivalued operator in a euclidean space is completely determined a linear relation can be written as a direct sum of three relations of different classes which are jordan relations completely singular relations and multishifts all three classes of relations are characterized in terms of the spectrum and their chain structure which leads to a generalization of the classical jordan canonical form

a dynamic subgradient based branch and bound procedure for set covering we discuss a branch and bound algorithm for set covering whose centerpiece is a new integrated upper bounding lower bounding procedure called dynamic subgradient optimization dynsgrad this new procedure applied to a lagrangean dual at every node of the search tree combines the standard subgradient method with primal and dual heuristics that interact to change the lagrange multipliers and tighten the upper and lower bounds fix variables and periodically restate the lagrangean itself extensive computational testing is reported as a stand alone heuristic dynsgrad performs significantly better than other procedures in terms of the quality of solutions obtainable with a certain computational effort when incorporated into a branch and bound algorithm dynsgrad considerably advances the state of the art in solving set covering problems

an alternative pseudoperipheral node finder for resequencing schemes many resequencing algorithms for reducing the bandwidth profile and wavefront of sparse symmetric matrices have been published in finite element applications the sparsity of a matrix is related to the nodal ordering of the finite element mesh some of the most successful algorithms which are based on graph theory require a pair of starting pseudoperipheral nodes these nodes located at nearly maximal distance apart are determined using heuristic schemes this paper presents an alternative pseadoperipheral node finder which is based on the algorithm developed by gibbs poole and stockmeyer this modified scheme is suitable for nodal reordering of finite meshes and provides more consistency in the effective selection of the starting nodes in problems where the selection becomes arbitrary due to the number of candidates for these starting nodes this case arises in particular for square meshes the modified scheme was implemented in gibbs poole stockmeyer gibbs king and sloan algorithms test problems of these modified algorithms include 1 everstine s 30 benchmark problems 2 sets of square rectangular and annular cylindrical finite element meshes with quadrilateral and triangular elements and 3 additional examples originating from mesh refinement schemes the results demonstrate that the modifications to the original algorithms contribute to the improvement of the reliability of all the resequencing algorithms tested herein for the nodal reordering of finite element meshes

bohl exponent for time varying linear differential algebraic equations we study stability of linear time varying differential algebraic equations daes the bohl exponent is introduced and finiteness of the bohl exponent is characterised the equivalence of exponential stability and a negative bohl exponent is shown and shift properties are derived we also show that the bohl exponent is invariant under the set of bohl transformations for the class of daes which possess a transition matrix introduced in this article the bohl exponent is exploited to characterise boundedness of solutions of a cauchy problem and robustness of exponential stability

an in depth stability analysis of nonuniform fdtd combined with novel local implicitization techniques this work focuses on efficient full wave solutions of multiscale electromagnetic problems in the time domain three local implicitization techniques are proposed and carefully analyzed in order to relax the traditional time step limit of the finite difference time domain fdtd method on a nonuniform staggered tensor product grid newmark cranknicolson cn and alternating direction implicit adi implicitization all of them are applied in preferable directions alike hybrid implicitexplicit hie methods as to limit the rank of the sparse linear systems both exponential and linear stability are rigorously investigated for arbitrary grid spacings and arbitrary inhomogeneous possibly lossy isotropic media numerical examples confirm the conservation of energy inside a cavity for a million iterations if the time step is chosen below the proposed relaxed limit apart from the theoretical contributions new accomplishments such as the development of the leapfrog alternating direction hybrid implicit explicit adhie fdtd method and a less stringent courant like time step limit for the conventional fully explicit fdtd method on a nonuniform grid have immediate practical applications a novel newmark beta fdtd method is proposed in 3d the leapfrog adhie fdtd method is for the first time proposed a courant like time step limit less stringent than the currently known limits is found for conventional nonuniform fdtd the stability of three local implicitization techniques is rigorously analyzed in the algebraic sense the stability of the proposed implicitization techniques is numerically validated by a cavity example

averaging for switched daes convergence partial averaging and stability abstract averaging is a useful technique to simplify the analysis of switched systems in this paper we present averaging results for the class of systems described by switched differential algebraic equations daes conditions on the consistency projectors are given which guarantee convergence towards a non switched averaged system a consequence of this result is the possibility to stabilize switched daes via fast switching we also study partial averaging in case the consistency projectors do not satisfy the conditions for convergence the averaged system is then still a switched system but is simpler than the original the practical interest of the theoretical averaging results is demonstrated through the analysis of the dynamics of a switched electrical circuit

the kalman yakubovich popov inequality for differential algebraic systems existence of nonpositive solutions abstract the kalman yakubovich popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a popov function on the imaginary axis to the solvability of a linear matrix inequality in this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential algebraic systems our conditions are given in terms of positivity of a modified popov function in the right complex half plane our results also apply to non controllable systems consequences of our results are bounded real and positive real lemmas for differential algebraic systems

a projection method for the uncapacitated facility location problem several algorithms already exist for solving the uncapacitated facility location problem the most efficient are based upon the solution of the strong linear programming relaxation the dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function this paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections the amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints wherem is the number of clients for comparison the underlying linear programming dual hasmn m n variables andmn n constraints wheren is the number of potential locations for the facilities the method is flexible as it can handle side constraints in particular when there is a duality gap the linear programming formulation can be strengthened by adding cuts numerical results for some classical test problems are included

a conjugate rosen s gradient projection method with global line search for piecewise linear concave optimization abstract the kelley cutting plane method is one of the methods commonly used to optimize the dual function in the lagrangian relaxation scheme usually the kelley cutting plane method uses the simplex method as the optimization engine it is well known that the simplex method leaves the current vertex follows an ascending edge and stops at the nearest vertex what would happen if one would continue the line search up to the best point instead as a possible answer we propose the face simplex method which freely explores the polyhedral surface by following the rosen s gradient projection combined with a global line search on the whole surface furthermore to avoid the zig zagging of the gradient projection we propose a conjugate gradient version of the face simplex method for our preliminary numerical tests we have implemented this method in matlab this implementation clearly outperforms basic matlab implementations of the simplex method in the case of state of the art simplex implementations in c or similar our matlab implementation is only competitive for the case of many cutting planes

the mobius transform effect in singular systems of differential equations abstract the main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of differential equations and the solutions of its proper m systems i e systems that appear after applying the generalized mobius transform to the pencil of the original singular system prime system firstly we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of its proper m systems moreover these solutions can be explicitly represented without resorting to any further processes of computations finally numerical examples are given to illustrate our theory