the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

some proximity and sensitivity results in quadratic integer programming we show that for any optimal solution bar z for a given separable quadratic integer programming problem there exist an optimal solution bar x for its continuous relaxation such that parallel bar z bar x parallel infty leqslant n delta a wheren is the number of variables and a is the largest absolute subdeterminant of the integer constraint matrixa also for any feasible solutionz which is not optimal for the separable quadratic integer programming problem there exists a feasible solution bar z having greater objective function value and with parallel bar z z parallel infty leqslant n delta a we further prove under some additional assumptions that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb respectively depends linearly on b b 1 finally the validity of all the results for nonseparable mixed integer quadratic programs is established the proximity results obtained in this paper are extensions of some of the results described in cook et al 1986 for linear integer programming

lower and upper bounds for the allocation problem and other nonlinear optimization problems we demonstrate the impossibility of strongly polynomial algorithms for the allocation problem in the comparison model and in the algebraic tree computation model that follow from lower bound results consequently there are no strongly polynomial algorithms for nonlinear concave separable optimization over a totally unimodular constraint matrix this is in contrast to the case when the objective is linear we present scaling based algorithms that use a greedy algorithm as a subroutine the algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem in that the complexity meets the lower bound derived from the comparison model for other extensions of the allocation problem the scaling based algorithms presented here are the fastest known these algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables

reachability indices of periodic positive systems via positive shift similarity the problem of the paper is the formulation of the concept of reachability indices of periodic linear positive systems a positive system is one in which the positive orthant is an invariant subset with respect to positive input signals the concept of shift similarity for periodic positive systems is defined and the associated cyclically augmented system is considered the results are the relation of the positive similarity of both representations the formulation of the reachability indices and a detailed example

analytical and formal classifications of quasi homogeneous foliations in c2 0 we prove a result of classification for germs of formal and convergent quasi homogeneous foliations in c 2 with fixed separatrix basically we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy

triangularizing matrix polynomials for an algebraically closed field f we show that any matrix polynomial p lambda in f lambda nbym n le m can be reduced to triangular form preserving the degree and the finite and infinite elementary divisors we also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi triangularizable with diagonal blocks of sizes 1 times 1 and 2 times 2 the proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors

convex separable optimization is not much harder than linear optimization the polynomiality of nonlinear separable convex concave optimization problems on linear constraints with a matrix with small subdeterminants and the polynomiality of such integer problems provided the inteter linear version of such problems ins polynomial is proven this paper presents a general purpose algorithm for converting procedures that solves linear programming problems the conversion is polynomial for constraint matrices with polynomially bounded subdeterminants among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix to integer separable convex programming an algorithm for finding a e accurate optimal continuous solution to the nonlinear problem that is polynomial in log 1 e and the input size and the largest subdeterminant of the constraint matrix is also presented these developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function the practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function only a polynomial number of function evaluations on a prespecified grid

structured perturbation of the brunovsky form a particular case we study the variation of the feedback invariants of a complex rectangular n x n m matrix when we make small additive perturbations on the last m columns in the particular case when the square n x n submatrix is similar to a block diagonal matrix with two blocks in the diagonal associated with the controllable part and the noncontrollable part of the pair a b respectively

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

the invariant polynomials degrees of the kronecker sum of two linear operators and additive theory let g be an abelian group let a and b be finite non empty subsets of g by a b we denote the set of all elements a b with a a and b b for c a b c a b is the cardinality of the set of pairs a b such that a b c we call c a b the multiplicity of c in a b let i be a positive integer we denote by i a b or briefly by i the cardinality of the set of the elements of a b that have multiplicity greater than or equal to i let f be a field let p be the characteristic of f in case of finite characteristic and if f has characteristic 0 let a and b be finite non empty subsets of f we will prove that for every l 1 min a b one has a 1 l lmin p a b l this statement on the multiplicities of the elements of a b generalizes cauchy davenport theorem in fact cauchy davenport is exactly inequality a for l 1 when f zp inequality a was proved in j m pollard j london math soc 8 1974 460 462 see also m b nathanson additive number theory inverse problems and the geometry of sumsets springer new york 1996

a preparation theorem for codimension one foliations after gluing foliated complex manifolds we derive a preparation like theorem for singularities of codimension one foliations and planar vector fields in the real or complex setting without computation we retrieve and improve results of levinson moser for functions dufour zhitomirskii for nondegenerate codimension one foliations proving in turn the analyticity str oyzyna y zo ladek for non degenerate planar vector fields and bruno ecalle for saddle node foliations in the plane

feedback invariants of matrices with prescribed rows abstract let a f n n b f n m where f is an arbitrary field in this paper we solve the problem of the existence of the matrix a b when its feedback invariants and some rows are prescribed

local realizations and local polynomial matrix representations of systems we define the local polynomial matrix representations of a controllable matrix pair a b with elements in an arbitrary field f and the local realizations of a nonsingular polynomial matrix whose elements are in f s with respect to a nonempty subset of specm f s we give different characterizations of these local concepts in particular when f c local realizations and left null pairs as defined in gohberg et al i gohberg m a kaashoek f van schagen partially specified matrices and operators classification completion applications bikhauser basel 1995 are closely related moreover global polynomial matrix representations and global realizations as defined in zaballa i zaballa controllability and hermite indices of matrix pairs int j control 68 1 1997 61 86 are particular cases of the same local concepts finally local wiener hopf factorization indices with respect to a nonempty subset of specm f s are defined

assignment of complete structural properties of linear systems via sensor selection for x ax bu the problem of structural assignment via sensor selection is to find an output equation y cx du such that the resulting system a b c d has the pre specified structural properties such as the finite and infinite zero structures as well as the invertibility properties in this paper we establish a set of necessary and sufficient conditions under which a complete set of system structural properties can be assigned and an explicit algorithm for constructing the required matrix pair c d

canonical form for the refined wiener hopf equivalence relation for nonsingular 3 3 polynomial matrices abstract we obtain a canonical form for the left 2 1 wiener hopf equivalence at infinity for polynomial matrices of order 3 this equivalence relation appears in a problem of structured perturbation when we consider polynomial matrix representations associated with controllable pairs

a new notion of equivalence for discrete time ar representations we present a new equivalence transformation termed divisor equivalence that has the property of preserving both the finite and the infinite elementary divisor structures of a square non singular polynomial matrix this equivalence relation extends the known notion of strict equivalence which dealt only with matrix pencils to the general polynomial matrix case it is proved that divisor equivalence characterizes in a closed form relation the equivalence classes of polynomial matrices that give rise to fundamentally equivalent discrete time auto regressive representations

a unified method for a class of convex separable nonlinear knapsack problems abstract in this paper a unified algorithm is proposed for solving a class of convex separable nonlinear knapsack problems which are characterized by positive marginal cost pmc and increasing marginal loss cost ratio imlcr by taking advantage of these two characteristics the proposed algorithm is applicable to the problem with equality or inequality constraints in contrast to the methods based on karush kuhn tucker kkt conditions our approach has linear computation complexity numerical results are reported to demonstrate the efficacy of the proposed algorithm for different problems

poles and zeros of matrices of rational functions abstract this expository paper considers the problem of defining poles and zeros with multiplicity including those at infinity for a matrix of rational functions over a field the underlying theme question for the paper is does the number of poles of a matrix equal the number of zeros and what structural meaning underlies such an assertion our approach is motivated by ideas from linear system theory and control engineering control theoretic ideas are not a prerequisite and we hope to encourage specialists in classical linear and commutative algebra to investigate this promising source of new algebra problems

feedback invariants of restrictions and quotients series connected systems abstract the problem of the feedback invariants of parallel and series connected systems is considered it is shown that this problem is closely related to the characterization of the feedback invariants of a system with prescribed restriction and or quotient to a controlled invariant subspace namely under the assumption of complete controllability of the quotient the relationship between the feedback invariants of a b and those of its restriction and quotient is investigated

parametrising the solutions of the generalised partial realisation problem through an atlas of coordinates given a finite nice sequence of matrices and a positive integer d the set of generalised partial realisations of of order d is a stratified smooth manifold an atlas of coordinate charts for each stratum is provided that locally parametrises it in particular the set of all generalised partial realisations of order d of any given nice sequence will be described

controllability and hermite indices of matrix pairs given a pair of matrices a b we can associate to each nice basis of the controllability subspace of a b a set of integers to be called the indices of the basis each set provides partial information about the similarity class of a the relationship between several sets of indices as well as the constraints that they impose on the invariant factors of a are investigated special attention is paid to the controllability and hermite indices of a given pair a b

structure smith macmillan form and coprime mfds of a rational matrix inside a region p ω abstract the structure of the smith macmillan form of a rational matrix t s inside a region p where is asymmetric with respect to the real axis subset of the finite complex plane c is determined algorithmic procedures based on elementary row and column operations over the euclidean ring r p s consisting of all rational functions with no poles in p are given coprimeness in p of a pair of rational matrices is studied in detail these results lead to constructive procedures for determining the coprime in p matrix fraction descriptions of t a

algorithms for the continuous nonlinear resource allocation problem new implementations and numerical studies patriksson 2008 provided a then up to date survey on the continuous separable differentiable and convex resource allocation problem with a single resource constraint since the publication of that paper the interest in the problem has grown several new applications have arisen where the problem at hand constitutes a subproblem and several new algorithms have been developed for its efficient solution this paper therefore serves three purposes first it provides an up to date extension of the survey of the literature of the field complementing the survey in patriksson 2008 with more then 20 books and articles second it contributes improvements of some of these algorithms in particular with an improvement of the pegging that is variable fixing process in the relaxation algorithm and an improved means to evaluate subsolutions third it numerically evaluates several relaxation primal and breakpoint dual algorithms incorporating a variety of pegging strategies as well as a quasi newton method our conclusion is that our modification of the relaxation algorithm performs the best at least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

on the existence of linear systems with prescribed invariants for system similarity we define the local wiener hopf controllability and hermite indices of nonsingular polynomial matrices and controllable matrix pairs and deduce that the local indices of matrix pairs are the local indices of their polynomial matrix representations we solve the problem of the existence of nonsingular polynomial matrices with prescribed invariant factors and local and global either hermite or wiener hopf indices finally we apply this result to finding necessary and sufficient conditions for the existence of a controllable pair a b with prescribed invariant factors for a and local and global either hermite or controllability indices

rank one perturbations of matrix pencils abstract we solve the problem of characterizing the kronecker structure of a matrix pencil obtained by a rank one perturbation of another matrix pencil the results hold over arbitrary fields

optimal flows in networks with multiple sources and sinks with applications to oil and gas lease investment programs in the classical maximal flow problem the objective is to maximize the supply to a single sink in a capacitated network in this paper we consider general capacitated networks with multiple sinks the objective is to optimize a general concave preference relation on the set of feasible supply vectors we show that an optimal solution can be obtained by a marginal allocation procedure an efficient implementation results in an adaptation of the augmenting path algorithm we also discuss an application of the procedure for an investment company that deals in oil and gas ventures

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

equivalence transformations of rational matrices and applications the known theories of transformations between polynomial matrices are extended to the case of rational matrices specifically equivalence between rational matrices having possibly different dimensions is defined and this has the property of preserving the zero structure of rational matrices in the region c some implications of this new equivalence transformation for linear system theory are also provided

on coprime rational function matrices abstract the concept of coprimeness of matrices with elements in a field of fractions is introduced we focus on the field of rational functions and define when two rational matrices are coprime with respect to different rings the definition of coprimeness at infinity of polynomial matrices is obtained as a particular case moreover the fact that coprimeness is a local property is proved and it is shown how coprimeness changes by performing mobius transformations

m convex function minimization by continuous relaxation approach proximity theorem and algorithm the concept of m convexity for functions in integer variables introduced by murota adv math 124 1996 pp 272 311 plays a primary role in the theory of discrete convex analysis in this paper we consider the problem of minimizing an m convex function which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points we propose a new approach for m convex function minimization based on continuous relaxation by establishing proximity theorems we develop a new algorithm based on continuous relaxation we apply the approach to some special cases of the separable convex quadratic resource allocation problem and the convex quadratic tree resource allocation problem to obtain faster algorithms

solving 0 1 knapsack problems based on amoeboid organism algorithm the 0 1 knapsack problem is an open issue in discrete optimization problems which plays an important role in real applications in this paper a new bio inspired model is proposed to solve this problem the proposed method has three main steps first the 0 1 knapsack problem is converted into a directed graph by the network converting algorithm then for the purpose of using the amoeboid organism model the longest path problem is transformed into the shortest path problem finally the shortest path problem can be well handled by the amoeboid organism algorithm numerical examples are given to illustrate the efficiency of the proposed model

on a homeomorphism between orbit spaces of linear systems and matrix polynomials abstract the orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact open topology is considered in the latter as a consequence the variation of the finite and left wiener hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described

infinite structure and exact model matching problem a geometric approach we first generalize to a b c d quadruples the geometric characterization of the infinite zero structure we then give a new geometric proof for a solvability condition of the exact model matching problem expressed in terms of this concept and previously established in the polynomial context by morse verghese and pernebo this emphasizes the approach independent character of the infinite zero structure

feedback invariants via output injection abstract the feedback equivalence classes of matrix pairs a kc b kw obtained by output injection of matrix k on a system a b c w are completely characterized working on arbitrary fields some related problems are also solved

lower bounds for computations with the floor operation a general lower bound technique is developed for computation trees with operations lfloor cdot rfloor and constants 0 1 for functions that have as their input a single n bit integer the technique applies to many natural functions such as perfect square root deciding if the square root of the input is integral or not computing the parity of lfloor log x rfloor etc the arguments are then extended to obtain the same lower bounds on the time complexity of any ram program with operations lfloor cdot rfloor that solves the problem another related result is described in a companion paper proc 29th ieee symposium on foundations of computer science 1988 and j assoc comput mach 1991 to appear

coordinatewise domain scaling algorithm for m convex function minimization we present a polynomial time scaling algorithm for the minimization of an m convex function m convex functions are nonlinear discrete functions with poly matroid structures which are being recognized as playing a fundamental role in tractable cases of discrete optimization the algorithm is applicable also to a variant of quasi m convex functions

smith mcmillan factorizations at infinity of rational matrix functions and their control interpretation in this paper the smith mcmillan factorizations at infinity of a transfer function are studied the non uniqueness of these factorizations is characterized via a multiplicative group of bicausal isomorphisms control interpretations are given in terms of system equivalence under some groups of transformations a characterization of the stabilizer of morse group at a b c is given for irreducible systems

feedback invariants of restrictions a polynomial approach the possible feedback invariants of a linear time invariant dynamical system represented by a pair of matrices a b are studied when the only information about the system is that it has an a b invariant subspace s whose restriction is an autonomous system and the quotient is controllable this problem is shown to be related to the problem of designing a control matrix b 1 for a series connected system to have prescribed feedback invariants

filters connecting isospectral quadratic systems abstract this paper concerns n n matrix polynomials of the form l m 2 d k we study isospectral pairs l l and show how to construct linear matrix functions f f called filters for which f l l f the ultimate objective is to design filters in such a way that l is diagonal where this is possible the main result of this paper is a constructive theorem demonstrating that two systems are isospectral if and only if there exists an associated family of coprime filters well established techniques are used in the construction of filters using standard pairs for matrix polynomials and structure preserving transformations for their linearizations in this paper attention is confined to systems with nonsingular leading coefficient m

wiener hopf factorization indices and infinite structure of rational matrices the relationship between the infinite structure and the wiener hopf factorization indices of polynomial matrix representations of controllable systems is investigated on the way local wiener hopf factorization indices must be defined in addition the finite and infinite structure of a rational matrix is characterized given its wiener hopf factorization indices

hermite indices and the action of the feedback group the controllability indices are a complete system of invariants for the feedback equivalence relation of controllable matrix pairs the hermite indices are invariant for the similarity of matrix pairs but they are not invariant by changing bases on the input space and performing state feedback the aim of this work is to partially characterize the hermite indices of a controllable matrix pair when transformations of the feedback group are performed on it

infinite elementary divisors of a matrix polynomial and implications a new characterization of infinite poles and zeros in terms of a natural extension of the idea of infinite elementary divisors of a matrix pencil to matrix polynomials is developed it is used to show that two previous descriptions of infinite frequency structure are equivalent subsequently a polynomial matrix transformation which simultaneously preserves the finite and infinite zero structure is presented

about strongly polynomial time algorithms for quadratic optimization over submodular constraints we present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints the main results are an o nm log n 2 m algorithm for the problemnetwork defined on a network onm arcs andn nodes an o n logn algorithm for thetree problem onn variables an o n logn algorithm for thenested problem and a linear time algorithm for thegeneralized upper bound problem these algorithms are the best known so far for these problems the status of the general problem and open questions are presented as well

analytic classification of a class of cuspidal foliations in c3 0 abstract in this article we study the analytic classification of certain types of quasi homogeneous cuspidal holomorphic foliations in c 3 0 via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation

invariant factor assignment under state feedback and output injection for a given system a b c necessary and sufficient conditions are given to assign invariant factors to the matrix of the system performing a state feedback and an output injection over an arbitrary field

a heuristic algorithm for a chance constrained stochastic program a chance constrained stochastic program is considered that arises from an application to college enrollments and in which the objective function is the expectation of a linear function of the random variables when these random variables are independent and normally distributed with mean and variance that are linear in the decision variables the deterministic equivalent of the problem is a nonconvex nonlinear knapsack problem the optimal solution to this problem is characterized and a greedy type heuristic algorithm that exploits this structure is employed computational results show that the algorithm performs well especially when the normal random variables are approximations of binomial random variables

novel theory for polynomial and rational matrices at infinity part 2 rational matrices the development reported in part 1 of this series is continued by presenting a unifying theory for rational matrices at infinity the cornerstone for the development is again the homogeneous form for polynomial and rational matrices it is shown that when a rational matrix is factorized in matrix fraction description mfd redundancy at infinity may be introduced this redundancy can be characterized precisely as the common divisor of the numerator and the denominator matrices at infinity moreover when the numerator and the denominator matrices are completely coprime the determinantal degree of the denominator is the mcmillan degree of the rational matrix

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

development of a hybrid dynamic programming approach for solving discrete nonlinear knapsack problems abstract a multiple choice knapsack problem can be formulated as a discrete nonlinear knapsack problem dnkp a powerful method for solving dnkp is the dynamic programming solution approach the use of this powerful approach however is limited since the growth of the number of decision variables and state variables requires an extensive computer memory storage and computational time in this paper we developed a hybrid algorithm for improving the computational efficiency of the dynamic programming when it is applied for solving the dnkp in the hybrid algorithm three routines of the imbedded state surrogate constraints and bounding scheme are incorporated for increasing the efficiency of this solution approach we then conducted an experimental study for comparing the computational efficiency of the hybrid algorithm with the imbedded state dynamic programming approach

on matrix polynomials with the same finite and infinite elementary divisors abstract a criterion is presented that characterizes when two matrix polynomials of any size rank and degree have the same finite and infinite elementary divisors this characterization inherits a coprimeness condition of the extended unimodular equivalence defined by pugh and shelton 17 in the set of real or complex matrix polynomials satisfying the constraint that the difference between the number of rows and columns is constant this extended unimodular equivalence is first generalized to matrices of any size with elements in any principal ideal domain

new upper bounds and exact methods for the knapsack sharing problem a new upper bound is proposed for the knapsack sharing problem a new dichotomous search based exact algorithm is proposed the new algorithm outperforms a most recent exact algorithm of the literature and the cplex solver v 12 in this paper we study the knapsack sharing problem ksp which is a variant of the well known np hard knapsack problem we investigate the use of a dichotomous search based exact method for solving the ksp such a method is based on decomposing the original problem into a series of minimizing and maximizing knapsack problems where each of them is embedded into a dichotomous search bounded by both lower and upper bounds throughout the interval search we introduce new upper bounds and incremental valid lower bounds one of these upper bounds can be viewed as an extended of dantzig upper bound whereas a series of valid lower bounds are obtained when solving a subset of knapsack problems finally the proposed algorithm is evaluated on a set of problem instances of the literature and other new harder ones the provided results are compared to those reached by the cplex solver and a more recent exact algorithm of the literature the computational part shows the dominance of the new exact method

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

the nonlinear knapsack problem algorithms and applications abstract we present a survey of algorithms and applications for the nonlinear knapsack problem or the nonlinear resource allocation problem in its most general form the nonlinear knapsack problem will be defined as a nonlinear optimization problem with just one constraint bounds on the variables and in some cases a set of specially structured constraints such as generalized upper bounds gubs this problem is encountered either directly or as a subproblem in a variety of applications including production planning financial modeling stratified sampling and capacity planning in manufacturing health care and computer networks by taking advantage of the special structure of the problem efficient solution methods can be developed problem classes addressed here include continuous and integer problems convex and nonconvex problems separable and nonseparable problems and problems with additional specially structured constraints

local wiener hopf factorization and indices over arbitrary fields abstract in this paper a generalization to arbitrary fields of the usual wiener hopf equivalence of complex valued rational matrix functions is given and the left local wiener hopf factorization indices defined in our previous work a amparan s marcaida i zaballa local realizations and local polynomial matrix representations of systems linear algebra appl 425 2007 757 775 are proved to form a complete system of invariants for this equivalence relation for the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices a amparan s marcaida i zaballa on the existence of linear systems with prescribed invariants for system similarity linear algebra appl 413 2006 510 533

disaggregation and resource allocation using convex knapsack problems with bounded variables the allocation of a specific amount of a given resource among competitive alternatives can often be modelled as a knapsack problem this model formulation is extremely efficient because it allows convex cost representation with bounded variables to be solved without great computational efforts practical applications of this problem abound in the fields of operations management finance manpower planning marketing etc in particular knapsack problems emerge in hierarchical planning systems when a first level of decisions need to be further allocated among specific activities which have been previously treated in an aggregate way in this paper we provide a recursive procedure to solve such problems the method differs from classical optimization algorithms of convex programming in that it determines at each iteration the optimal value of at least one variable applications and computational results are presented

the greedy procedure for resource allocation problems necessary and sufficient conditions for optimality in many resource allocation problems the objective is to allocate discrete resource units to a set of activities so as to maximize a concave objective function subject to upper bounds on the total amounts allotted to certain groups of activities if the constraints determine a polymatroid and the objective is linear it is well known that the greedy procedure results in an optimal solution in this paper we extend this result to objectives that are weakly concave a property generalizing separable concavity we exhibit large classes of models for which the set of feasible solutions is a polymatroid and for which efficient implementations of the greedy procedure can be given

a refined wiener hopf equivalence relation for polynomial matrices abstract we introduce an equivalence relation which is finer than the left wiener hopf equivalence at infinity for polynomial matrices and we obtain discrete invariants and a reduced form for this equivalence relation