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

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

global attractivity in the recursive sequence xn 1 α βxn γ xn 1 our aim in this paper is to investigate the global attractivity of the recursive sequencex n email protected bx email protected n 1 n 0 1 where a 0 b c 0 we show that one positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

global attractivity in a rational recursive sequence our aim in this paper is to investigate the periodic character invariant intervals and the global attractivity of all positive solutions of the nonlinear difference equationx n email protected email protected email protected n 1 n 0 1 where a 0 b c 0 we show that the positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

on the positive solutions of the difference equation xn 1 xn 1 1 x nxn 1 the positive solutions of the difference equation x n 1 x n 1 1 x nx n 1 n 0 1 2 are investigated where x 1 and x 0 are the positive real numbers

a note on the periodic cycle of xn 2 1 xn 1 xn the behaviour of solutions of the difference equation x n 1 a n b nx nx n 1 n 0 1 are investigated where the sequences a n and b n are periodic

design of a degree constrained minimal spanning tree with unreliable links and node outage costs abstract the degree constrained minimal spanning tree dcmst problem with unreliable links and node outage costs consists of finding links in a network to connect a set of terminal nodes to a central node while minimizing the expected annual expenditure the number of ports available on each terminal node limits the number of incident links the degree constraint each terminal node in the network has an associated node outage cost which is the economic cost incurred by the network user whenever that node is disabled due to failure of a link we formulate this problem as an integer programming problem and present a lagrangian relaxation method which for each choice of lagrangian multipliers provides a lower bound for the optimal objective function value a subgradient optimization method is used to search for multipliers which yield good lower bounds a branch exchange heuristic procedure makes modifications to each infeasible solution of the lagrangian relaxation in order to find good feasible solutions the quality of these heuristic solutions is estimated using the best obtained lower bounds experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem

efficient preconditioners for optimality systems arising in connection with inverse problems this paper is devoted to the numerical treatment of linear optimality systems os that arise in connection with inverse problems for partial differential equations if such inverse problems are regularized by tikhonov regularization then it follows from standard theory that the associated os is well posed provided that the regularization parameter alpha is positive and that the involved state equation satisfies suitable assumptions we explain and analyze how certain mapping properties of the operators appearing in the os can be employed to define efficient preconditioners for finite element fe approximations of such systems the key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h used in the fe discretization and increases only moderately as alpha rightarrow0 more specifically if the stopping criterion for the iteration process is defined in terms of the associated energy norm then the number of iterations required in the severely ill posed case cannot grow faster than o ln alpha 2 our analysis is based on a careful study of the operators involved which yields the distribution of the eigenvalues of the preconditioned os finally the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by borzi kunisch and kwak siam j control optim 41 2003 pp 1477 1497 and an inverse problem arising in connection with electrocardiography nielsen cai and lysaker math biosci 210 2007 pp 523 553

a restricted lagrangean approach to the traveling salesman problem we describe an algorithm for the asymmetric traveling salesman problem tsp using a new restricted lagrangean relaxation based on the assignment problem ap the lagrange multipliers are constrained so as to guarantee the continued optimality of the initial ap solution thus eliminating the need for repeatedly solving ap in the process of computing multipliers we give several polynomially bounded procedures for generating valid inequalities and taking them into the lagrangean function with a positive multiplier without violating the constraints so as to strengthen the current lower bound upper bounds are generated by a fast tour building heuristic when the bound strengthening techniques are exhausted without matching the upper with the lower bound we branch by using two different rules according to the situation the usual subtour breaking disjunction and a new disjunction based on conditional bounds we discuss computational experience on 120 randomly generated asymmetric tsp s with up to 325 cities the maximum time used for any single problem being 82 seconds this is a considerable improvement upon earlier methods though the algorithm discussed here is for the asymmetric tsp the approach can be adapted to the symmetric tsp by using the 2 matching problem instead of ap

on the periodic cycle of xn 1 an bnxn cnxn 1 the behaviour of solutions of the difference equation x n 1 a n b nx n c nx n 1 n 0 1 2 are investigated where the sequence a n b n and c n are periodic

a multiperiod planning model for the capacitated minimal spanning tree problem the multiperiod capacitated minimal spanning tree mcmst problem consists of scheduling the installation of links in a network so as to connect a set of terminal nodes s 2 3 n to a central node node 1 with minimal present value of expenditures where link capacities limit the number of terminal nodes sharing a link some of the terminal nodes are active at the beginning of the planning horizon while others are activated over time we formulate this problem as an integer programming problem a branch exchange heuristic procedure for solving the problem is presented we also present a lagrangian relaxation method to find a lower bound for the optimal objective function value this lower bound may be used to estimate the quality of the solution given by the branch exchange heuristic experimental results over a wide range of problem structures show that the branch exchange heuristic method yields verifiably good solutions to this problem

order optimal preconditioners for implicit runge kutta schemes applied to parabolic pdes in this paper we show that standard preconditioners for parabolic pdes discretized by implicit euler or crank nicolson schemes can be reused for higher order fully implicit runge kutta time discretization schemes we prove that the suggested block diagonal preconditioners are order optimal for a stable and irreducible runge kutta schemes with invertible coefficient matrices the theoretical investigations are confirmed by numerical experiments

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

on the recursive sequence xn axn 1 bxn 2c dxn 1xn 2 in this paper we study the invariant intervals the global attractivity of the equilibrium points and the asymptotic behavior of the solutions of the difference equationx n ax n 1 bx n 2c dx n 1x n 2 n 1 2 where a 0 b c d 0

global asymptotic stability in a rational recursive sequence in this paper we study the global stability of the difference equationx n a bx n 1 cx n 1 2d x n 2 n 1 2 where a b 0 and c d 0 we show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters and every positive solution of the equation in the basin exponentially converges to the attractor

global attractivity of the difference equation xn 1 α xn k xn in this paper we investigate the global stability of all negative solutions of the difference equationx n email protected x n kx n n 0 1 where a 1 is an integer and the initial conditions x k x 0 are arbitrary real numbers we show that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

global attractivity for a class of higher order nonlinear difference equations our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationx n 1 a bx email protected k i 0b ix n i n 0 1 where a b b email protected 0 a b 0 b k email protected 0 email protected 1 2 and the initial conditions x k x 0 are arbitrary real numbers we show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

multilevel iteration for mixed finite element systems with penalty in this paper the authors consider the solution of the discrete systems that arises when a mixed finite element approach is used to approximate the solution of second order elliptic boundary value problems by the introduction of a penalty parameter these equations can be approximated by the solution of a symmetric and positive definite penalty system on the velocity subspace iterative procedures are developed and analyzed for this penalty system based on the hierarchical basis approach as well as on the standard multigrid approach finally numerical experiments are presented that illustrate the convergence behavior suggested by the theory

a 2n constraint formulation for the capacitated minimal spanning tree problem in this paper we present a new formulation for the capacitated minimal spanning tree cmst problem one advantage of the new formulation is that it is more compact in the number of constraints than a well known formulation additionally we show that the linear programming relaxation of both formulations produces optimal solutions with the same cost we present a brief discussion concerning valid inequalities for the cmst which are directly derived from the new formulation we show that some of the new inequalities are not dominated by some sets of facet inducing inequalities for the cmst we derive some lagrangian relaxation based methods from the new formulation and present computational evidence showing that reasonable improvements on the original linear programming bounds can be obtained if these methods are strengthened by the use of cutting planes the reported computational results indicate that one of the methods presented in this paper dominates in most of the cases the previous best methods reported in the literature the most significant improvements are obtained in the instances with the root in the corner

augmented lagrangean based algorithms for centralized network design capacitated spanning tree problems appear frequently as fundamental problems in many communication network design problems an integer programming formulation and a new set of valid inequalities are presented for the linear characterization of the problem a combination of a subgradient optimization procedure and an augmented lagrangean based procedure is used to generate tight lower bounds the procedure begins with an explicit representation of a subset of the constraints and the corresponding lagrangean problem is solved the solution is examined in order to identify implicit constraints that are violated those are added to the lagrangean problem forming an expanded problem and an efficient dual ascent procedure is then applied when no further improvement is possible through this procedure a subgradient optimization procedure is invoked in order to further tighten the lower bound value an exchange heuristic is applied to the nonfeasible lagrangean solution in an attempt to generate good feasible solutions to the problem the procedure has been tested and has generated bounds that are significantly better than ones obtained through previously published procedures

algorithms for the multi resource generalized assignment problem the multi resource generalized assignment problem is encountered when a set of tasks have to be assigned to a set of agents in a way that permits assignment of multiple tasks to an agent subject to the availability of a set of multiple resources consumed by that agent this problem differs from the generalized assignment problem in that an agent consumes not just one but a variety of resources in performing the tasks assigned to him this paper develops effective solution procedures for the multi resource generalized assignment problem various relaxations of the problem are studied and theoretical relations among these relaxations are pointed out rules for reducing problem size are discussed and are shown to be effective through computational experiments heuristic solution procedures and an efficient branch and bound procedure are developed results of computational experiments testing these procedures are reported

a domain embedding preconditioner for the lagrange multiplier systems finite element approximations for the dirichlet problem associated to a second order elliptic differential equation are studied the purpose of this paper is to discuss domain embedding preconditioners for discrete systems the essential boundary condition on the interior interface is removed by introducing lagrange multipliers the associated discrete system with a saddle point structure is preconditioned by a block diagonal preconditioner the main contribution of this paper is to propose a new operator constructed from the h div inner product for the block of the preconditioner corresponding to the multipliers

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

on the difference equation xn 1 a αxn αxn 1 αxn k 2 xn k 1 in this paper we establish invariant relations boundedness persistence strictly oscillatory and semi cycles for the positive solutions of the difference equationx n email protected x n x n 1 x n k 2 x n k 1 n k 1 k email protected 0 email protected 0 email protected 1 2

dynamic behavior of a recursive sequence in this paper the global asymptotic stability the global attractivity the boundedness character the periodic nature and chaotic behavior of solutions of the difference equationx n 1 a x n 1x n n 0 1 are investigated where a r is a real number and the initial conditions x 1 x 0 are arbitrary real numbers as might be expected the six cases a 3 give rise to substantially different dynamic behavior

heuristics with constant error guarantees for the design of tree networks a tree network is a collection of trees rooted at a common central node several types of network design problems can be viewed as requiring the formation of a spanning tree network of minimum length subject to a bound on the sum of weights on the nodes of any component tree such problems are np complete and experience has shown that only small examples can be solved to optimality this paper describes an efficient heuristic algorithm based on partitioning of a traveling salesman tour when all the nodes have a unit weight and the bound is k the heuristic finds a solution whose cost is at most 3 2 k times the minimum in the general case the error bound is 4

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

uniform preconditioners for the time dependent stokes problem implicit time stepping procedures for the time dependent stokes problem lead to stationary singular perturbation problems at each time step these singular perturbation problems are systems of saddle point type which formally approach a mixed formulation of the poisson equation as the time step tends to zero preconditioners for discrete analogous of these systems are discussed the preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization the construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system

dynamics of a rational difference equation in this paper we investigate the periodic character invariant intervals oscillation and global stability of all positive solutions of the equationx n 1 px n x n kq x n k n 0 1 where the parameters p and q and the initial conditions x k x 0 are nonnegative real numbers as might be expected the two cases p q give rise to different dynamic behaviors in particular our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2001

global attractivity in a recursive sequence in this paper we study the invariant intervals the globally attractivity of the two equilibrium points and the oscillatory behavior of the solutions of the difference equationx n ax n 1 bx n 2c x n 2 n 1 2 where a b c 0