a relative error bound for balanced stochastic truncation recently several methods have appeared for the approximation of power spectra notably balanced stochastic truncation bst it is shown that bst satisfies a relative error bound approximately twice the bound for the relative error method rem proper this offers a quantitative basis for the observation that bst and rem produce similar reduced order models balanced stochastic truncation can therefore be interpreted as providing a computationally simple algorithm for relative error approximation

parametric h sub spl infin loopshaping and weighted mixed sensitivity minimization generalized h sub spl infin loopshaping is developed for a particular weighted mixed sensitivity minimization task a free parameter is introduced to tradeoff minimizations of sensitivity and complementary sensitivity in the h sub spl infin performance index a quadratically convergent algorithm is derived for computing the optimal performance index and a state space solution is obtained for suboptimal controllers these results are applicable to stabilization of uncertain systems where normalized coprime factors involve parametrically weighted h sub spl infin norm bounded perturbations controller reduction is also investigated for the parametric h sub spl infin loopshaping robust stability condition for using the reduced order controller is established and an estimate for the associated h sub spl infin performance is obtained similar results are derived for normalized linear quadratic gaussian lqg control

an h minimax approach to the design of robust control systems abstract a realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices a solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone

a connection between normalized coprime factorizations and linear quadratic regulator theory given a transfer matrix described by a minimal state space triple a method is given for computing state space realizations for the numerator and denominator of a normalized stable right coprime factorization for the transfer matrix the method involves the solution of an algebraic riccati equation it allows the use of existing computational state space algorithms in finding normalized stable right coprime factorizations and avoids explicit calculations of spectral factors

precedence theorems and dynamic programming for the single machine weighted tardiness problem abstract we tackle precedence constrained sequencing on a single machine in order to minimize total weighted tardiness classic dynamic programming dp methods for this problem are limited in performance due to excessive memory requirements particularly when the precedence network is not sufficiently dense over the last decades a number of precedence theorems have been proposed which distinguish dominant precedence constraints for a job pool that is initially without precedence relation in this paper we connect and extend the findings of the foregoing two strands of literature we develop a framework for applying the precedence theorems to the precedence constrained problem to tighten the search space and we propose an exact dp algorithm that utilizes a new efficient memory management technique our procedure outperforms the state of the art algorithm for instances with medium to high network density we also empirically verify the computational gain of using different sets of precedence theorems

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

controller reduction with error performance continuous and discrete time cases this paper is concerned with the problem of controller reduction for linear systems both continuous and discrete time cases are considered with necessary and sufficient conditions obtained for the existence of desired reduced order controllers in solving this problem two approaches are presented the first approach is based on the projection lemma where the admissible controllers can be parameterized after a set of conditions are satisfied and the second one directly incorporates the controller matrices to be determined into a set of conditions by introducing new techniques and thus no parameterization procedure is needed these necessary and sufficient conditions are formulated in terms of linear matrix inequalities lmis plus some equality constraints since these conditions are not convex the cone complementarity linearization ccl idea is exploited to cast them into sequential minimization problems subject to lmi constraints which can be readily solved by standard numerical software a num

on the structure of all minimum cuts in a network and applications this paper presents a characterization of all minimum cuts separating a source from a sink in a network a binary relation is associated with any maximum flow in this network and minimum cuts are identified with closures for this relation as a consequence finding all minimum cuts reduces to a straightforward enumeration applications of this results arise in sensitivity and parametric analyses of networks the vertex packing and maximum closure problems in unconstrained pseudo boolean optimization and project selection as well as in other areas of application of minimum cuts

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

on the approximability of single machine scheduling with precedence constraints we consider the single machine scheduling problem to minimize the weighted sum of completion times under precedence constraints in a series of recent papers it was established that this scheduling problem is a special case of minimum weighted vertex cover in this paper we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in the dimension theory of partial orders exploiting this relationship allows us to present a framework for obtaining 2 2 f approximation algorithms provided that the set of precedence constraints has fractional dimension of at most f our approach yields the best known approximation ratios for all previously considered special classes of precedence constraints and it provides the first results for bounded degree and orders of interval dimension 2 on the negative side we show that the addressed problem remains np hard even when restricted to the special case of interval orders furthermore we prove that the general problem if a fixed cost present in all feasible schedules is ignored becomes as hard to approximate as vertex cover we conclude by giving the first inapproximability result for this problem showing under a widely believed assumption that it does not admit a polynomial time approximation scheme

on the robust stability of linear time invariant plants with unstructured uncertainty we consider general families of linear time invariant plants described by a nominal plant model with unstructured uncertainty we show under a rather weak assumption that when a family of systems is not robustly stabilizable by any linear time invariant feedback controller then no nonlinear time varying controller can robustly stabilize the given family

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

stronger lagrangian bounds by use of slack variables applications to machine scheduling problems lagrangian relaxation is a powerful bounding technique that has been applied successfully to manynp hard combinatorial optimization problems the basic idea is to see annp hard problem as an easy to solve problem complicated by a number of nasty side constraints we show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds the trick is widely applicable but we focus on a broad class of machine scheduling problems for which it is particularly useful we provide promising computational results for three problems belonging to this class for which lagrangian bounds have appeared in the literature the single machine problem of minimizing total weighted completion time subject to precedence constraints the two machine flow shop problem of minimizing total completion time and the single machine problem of minimizing total weighted tardiness

multiplicative error bound for balanced stochastic truncation model reduction the multiplicative error for the balanced stochastic truncation model reduction method is discussed and an h sub infinity norm error bound is presented the result together with a previous result of the authors syst contr lett vol 14 p 307 14 1990 establishes that the same error bound applies to the relative error and the multiplicative error

preemptive and non preemptive generalized min sum set cover in the non preemptive generalized min sum set cover problem we are given n ground elements and a collection of sets mathcal s s 1 s 2 ldots s m where each set s i in 2 n has a positive requirement kappa s i that has to be fulfilled we would like to order all elements to minimize the total weighted cover time of all sets the cover time of a set s i is defined as the first index j in the ordering such that the first j elements in the ordering contain kappa s i elements in s i this problem was introduced by azar et al 2009 with interesting motivations in web page ranking and broadcast scheduling for this problem constant approximations are known by bansal et al 2010 and skutella and williamson oper res lett 39 6 433 436 2011 we study the version where preemption is allowed the difference is that elements can be fractionally scheduled and a set s is covered in the moment when kappa s amount of elements in s are scheduled we give a 2 approximation for this preemptive problem our linear programming relaxation and analysis are completely different from the aforementioned previous works we also show that any preemptive solution can be transformed into a non preemptive one by losing a factor of 6 2 in the objective function as a byproduct we obtain an improved 12 4 approximation for the non preemptive problem

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

an exact algorithm for the precedence constrained single machine scheduling problem abstract this study proposes an efficient exact algorithm for the precedence constrained single machine scheduling problem to minimize total job completion cost where machine idle time is forbidden the proposed algorithm is based on the ssdp successive sublimation dynamic programming method and is an extension of the authors previous algorithms for the problem without precedence constraints in this method a lower bound is computed by solving a lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence constrained total weighted tardiness and total weighted earliness tardiness problems and most instances with 100 jobs of the former problem

a system theoretic appropriate realization of the empty matrix concept an algebraic realization of the empty matrix concept that is appropriate for system theoretic applications is proposed the utility of the realization of the empty matrix concept and the deficiencies of the current matlab realization of this concept are demonstrated using examples these examples fully delineate how the empty matrix concept can be utilized to transparently handle static and for single vector input single vector output systems within the more general context of dynamic two vector input two vector output systems

controller reduction for linear parameter varying systems with a priori bounds in this paper we develop a controller reduction procedure for linear parameter varying lpv systems the method uses synthesis riccati inequalities for the normalized robust stabilization problem as a basis for the approximation the technique provides a priori error bounds which are used to obtain closed loop stability conditions and performance degradation level we also generalize the relative model reduction method to lpv systems and give an energy interpretation to the controller reduction procedure to illustrate the method a reduced order controller is synthesized and tested on a nonlinear missile model

fixed points of generalized contractive multi valued mappings in a recent paper n mizoguchi and w takahashi gave a positive answer to the conjecture of s reich concerning the existence of fixed points of multi valued mappings that satisfy a certain contractive condition in this paper we provide an alternative and somewhat more straightforward proof for the theorem of mizoguchi and takahashi also the problems associated with fixed points of weakly contractive multi valued mappings are studied finally we make a few comments that improve other results from their paper j math anal appl 141 1989 177 188

single machine scheduling with scenarios in the field of robust optimization the goal is to provide solutions to combinatorial problems that hedge against variations of the numerical parameters this constitutes an effort to design algorithms that are applicable in the presence of uncertainty in the definition of the instance we study the single machine scheduling problem with the objective of minimizing the weighted sum of completion times we model uncertainty by replacing the vector of numerical values in the description of the instance by a set of possible vectors called scenarios the goal is to find a schedule of minimum value in the worst case scenario we first show that the general problem cannot be approximated within o log 1 en for any e 0 unless np has quasi polynomial algorithms we then study more tractable special cases and obtain a linear program lp based 2 approximation algorithm for the unweighted case we show that our analysis is tight by providing a matching lower bound on the integrality gap of the lp moreover we prove that the unweighted version is np hard to approximate within a factor less than 6 5 we conclude by presenting a polynomial time algorithm based on dynamic programming for the case when the number of scenarios and the values of the instance are bounded by some constant

endpoints of set valued contractions in metric spaces abstract suppose x d be a complete metric space and suppose f x c b x be a set valued map satisfies h f x f y d x y for each x y x where 0 0 is upper semicontinuous t t for each t 0 and satisfies lim inf t t t 0 then f has a unique endpoint if and only if f has the approximate endpoint property

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

optimal h multivariable robust controllers and the relationship to lqg design problems a new solution for the multivariable minimax optimal robust design problem is presented the relationship between lqg and h design methods is considered and the h solution is obtained from the solution of a related lqg problem the approach provides insights into the relationship between the h and lqg design philosophies the h solution involves the same type of steps which are involved in the polynomial based solutions of the lqg problem namely spectral factorization and the solution of linear diophantine equations a simplified design procedure is described which requires only the solution of a standard eigenvalue eigenvector problem

a lagrangean based branch and bound algorithm for single machine sequencing with precedence constraints to minimize total weighted completion time the single machine sequencing problem is considered in which there are precedence constraints on the jobs the objective is to minimize the sum of weighted completion times a lower bound is obtained by successively performing a lagrangean relaxation of appropriate constraints each lagrange multiplier is chosen to provide the maximum increment to the lower bound subject to retaining the nonnegativity of the coefficients of the variables when no further suitable constraints can be introduced into the lagrangean function the variables having zero cost coefficient are used to obtain a feasible sequence which provides an upper bound the gap between the lower and upper bound is reduced by removing some constraints from the lagrangean function and replacing them with others this lower bounding procedure is used in a branch and bound algorithm computational results indicate that the algorithm can satisfactorily solve problems with up to 100 jobs

single machine scheduling polyhedra with precedence constraints we consider nonpreemptive single machine scheduling subject to precedence constraints we define feasible schedules by the vector of the job completion times and study the structure of the convex hull of all feasible schedules called the scheduling polyhedron p we derive classes of valid inequalities for p and necessary and sufficient conditions under which they are facet inducing our main result is a complete description of the minimal linear system defining p when the precedence constraints are series parallel moreover this system consists of two classes of facet inducing inequalities associated with series and parallel compositions respectively if the precedence constraints are not series parallel we present another class of facet inducing inequalities for p associated with induced z subgraphs we also show that the convex hull of all preemptive feasible schedules is identical to p if and only if the precedence constraints form an out forest

connection of multiplicative relative perturbation in coprime factors and gap metric uncertainty in this paper it is shown that a linear uncertain system described by a certain l multiplicative or relative perturbation in its coprime factors that are not necessarily normalized is the same as the one described by a gap or l gap metric ball hence all the stability robustness results for gap or l gap metric uncertainty carry over to this type of coprime factor perturbation uncertain systems described by h multi plicative or relative perturbations in coprime factors are also studied in this paper necessary and su cient conditions for robust stability of a feedback system with coprime factors of both the plant and the controller subject to simultaneous h multiplicative or relative perturbations are obtained

single machine scheduling with precedence constraints we discuss the problem of sequencing precedence constrained jobs on a single machine to minimize the average weighted completion time this problem has attracted much attention in the mathematical programming community since sidney s pioneering work in 1975 sidney j b 1975 decomposition algorithms for single machine scheduling with precedence relations and deferral costs operations research23 283 298 we look at the problem from a polyhedral perspective and uncover a relation between sidney s decomposition theorem and different linear programming relaxations more specifically we present a generalization of sidney s result which particularly allows us to reason that virtually all known 2 approximation algorithms are consistent with his decomposition moreover we establish a connection between the single machine scheduling problem and the vertex cover problem indeed in the special case of series parallel precedence constraints we prove that the sequencing problem can be seen as a special case of the vertex cover problem we also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem finally we give a 3 2 approximation algorithm for two dimensional partial orders and we also provide a characterization of the active inequalities of a linear programming relaxation in completion time variables

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

balanced stochastic realizations abstract the balanced stochastic realization is introduced as a balanced solution to the continuous time positive real equations and the dual positive real equations the structure of the associated balanced spectral factors is derived the main result being a product decomposition of the spectral factors the properties of the stochastic model reduction technique of balanced stochastic truncation are considered in particular it is shown that balanced stochastic truncation preserves the right half plane zeros of the spectral factors leading to a transformation approach to the approximation of non minimum phase stochastic systems

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

normalized h sub spl infin controller reduction with a priori error bounds this paper describes an approach to the reduction of controllers for the normalized coprime factor robustness problem as well as the normalized h sub spl infin problem it is shown that a relative error approximation of a coprime factor representation of any suboptimal controller leads to a stability guarantee and an upper bound on the performance degradation when the reduced order controller is implemented when the approximation is performed on the controller generator guaranteed a priori stability and performance bounds are obtained in terms of the synthesis riccati equation solutions of the normalized h sub spl infin control problems

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

endpoints and fixed points of set valued contractions in cone metric spaces abstract in this paper we introduce the concept of set valued contractions in cone metric spaces and establish new endpoint and fixed point theorems for such contractions the paper includes examples

existence and uniqueness of endpoints of closed set valued asymptotic contractions in metric spaces abstract investigations of problems of set valued asymptotic fixed point theory of set valued dynamic systems are initiated the concepts of the set valued asymptotic contractions of set valued dynamic systems t m 2 m in metric space m are introduced and conditions guaranteeing the existence and uniqueness of endpoints v m of set valued dynamic systems of set valued asymptotic contractions t m 2 m and convergence to v of all generalized sequences of iterations of t are established examples and remarks show a fundamental difference between our results and the well known ones

normalised coprime factorizations for nonstrictly proper systems some earlier results of d g meyer and g f franklin see ibid vol ac 32 pp 227 8 march 1987 are extended and a procedure is derived for obtaining directly a normalized right coprime factorization of a system that may not be strictly proper

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

near optimal solutions and large integrality gaps for almost all instances of single machine precedence constrained scheduling we consider the problem of minimizing the weighted sum of completion times on a single machine subject to bipartite precedence constraints in which all minimal jobs have unit processing time and zero weight and all maximal jobs have zero processing time and unit weight for various probability distributions over these instances including the uniform distribution we show several almost all type results first we show that almost all instances are prime with respect to a well studied decomposition for this scheduling problem second we show that for almost all instances every feasible schedule is arbitrarily close to optimal finally for almost all instances we give a lower bound on the integrality gap of various linear programming relaxations of this problem

model reduction with relative multiplicative error bounds and relations to controller reduction balanced model reduction with a priori relative multiplicative error bounds in l sub spl infin norm is studied inverse weighted balanced truncation iwbt is proposed that is a variant of the enns algorithm it is shown that iwbt is equivalent to the balanced stochastic truncation bst this equivalence leads to the extension of an improved relative multiplicative error bound for bst obtained by wang and safonov to general nonsquare transfer matrices natural relations between iwbt and observer based controller reduction are investigated which motivate the simultaneous reduction of the plant and controller robust stability conditions for simultaneous reduction of the plant and the controller are established for the closed loop system consisting of full order plant and reduced order controller as a by product robust stability and stabilization problems are solved for general coprime factor plant descriptions with relative multiplicative h sub spl infin norm bounded uncertainties similar results are obtained for general coprime factors of the feedback controller involving relative multiplicative h sub spl infin norm bounded uncertainties

feedback stability under simultaneous gap metric uncertainties in plant and controller abstract the stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric a fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

vertex cover in graphs with locally few colors erdos et al defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex for any d 2 there are graphs with arbitrarily large chromatic number that can be colored so that i no vertex neighborhood contains more than d different colors bounded local colorability and ii adjacent vertices from two color classes induce a complete bipartite graph biclique coloring we investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given this generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number assuming the unique game conjecture ugc we provide a tight characterization we prove that it is ugc hard to improve the approximation ratio of 2 2 d 1 on d 1 locally but not necessarily biclique colorable graphs a matching upper bound is also provided vice versa when properties i and ii hold we present a randomized algorithm with approximation ratio of 2 w 1 lnln dln d this matches known inapproximability results for the special case of bounded degree graphs moreover we show that when both the above two properties i and ii hold the obtained result finds a natural application in a classical scheduling problem namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time denoted as 1 prec w jc j in standard scheduling notation in a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs g p of incomparable pairs defined in the dimension theory of partial orders we show that g p satisfies properties i and ii where d 1 is the maximum number of predecessors or successors of each job

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

endpoint theory for set valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform spaces in uniform spaces inspired by ideas of banach tarafdar and yuan we introduce the concepts of generalized pseudodistances and generalized gauge maps for set valued dynamic systems we define various nonlinear asymptotic contractions and contractions with respect to these pseudodistances and gauges provide conditions on the iterates of these set valued dynamic systems and present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints stationary points of these set valued dynamic systems and conditions that each generalized sequence of iterations in particular each dynamic process converges and the limit of a generalized sequence of iterations is an endpoint the definitions the results and the method are new for set valued dynamic systems in uniform locally convex and metric spaces and even for single valued maps the paper includes a number of various examples which show a fundamental difference between our results and those existing in the literature

endpoints of multi valued generalized weak contraction mappings abstract let x d be a complete metric space and let t x p cl bd x be a multi valued generalized weak contraction mapping then t has a unique endpoint if and only if t has the approximate endpoint property our results extend previous results given by ciric 1971 15 nadler 1969 11 daffer and kaneko 1995 9 and amini harandi 2010 8

a controller degree bound for h 0e control problems of the second kind this paper is a continuation of our work on mathcal h infty optimal control problems which may be embedded in the linear fractional configuration of fig 1 in two previous articles 19 20 a controller degree bound was established for problems in which both p 12 s and p 21 s are square problems of the first kind if the mcmillan degree of p s is n it was shown that there exist mathcal h infty optimal controllers with mcmillan degree no greater than n 1 here we switch our attention to problems of the second kind that is we allow p 12 s to have more rows than columns with p 21 s square or alternatively we allow p 21 s to have more columns than rows with p 12 s square our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change in addition to the controller degree bound our analysis suggests a number of modifications which are easily made to cur

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs