aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

consistent and mimetic discretizations in general relativity a discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level such discretizations have been found useful in continuum mechanics and in electromagnetism we have recently introduced a new technique for discretizing constrained theories the technique yields discretizations that are consistent in the sense that the constraints and evolution equations can be solved simultaneously but it cannot be considered mimetic since it achieves consistency by determining the lagrange multipliers in this paper we would like to show that when applied to general relativity linearized around a minkowski background the technique yields a discretization that is mimetic in the traditional sense of the word we show this using the traditional metric variables and also the ashtekar new variables but in the latter case we restrict ourselves to the euclidean case we also argue that there appear to exist conceptual difficulties to the construction of a mimetic formulation of the full einstein equations and suggest that the new discretization scheme can provide an alternative that is nevertheless close in spirit to the traditional mimetic formulations

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

fatcop a fault tolerant condor pvm mixed integer programming solver we describe fatcop a new parallel mixed integer program solver written in pvm the implementation uses the condor resource management system to provide a virtual machine composed of otherwise idle computers the solver differs from previous parallel branch and bound codes by implementing a general purpose parallel mixed integer programming algorithm in an opportunistic multiple processor environment as opposed to a conventional dedicated environment it shows how to make effective use of resources as they become available while ensuring the program tolerates resource retreat the solver performs well on test problems arising from real applications and is particularly useful for solving long running hard mixed integer programming problems

a framework for constructing general integer problems with well determined duality gaps abstract the paper is concerned with constructing general integer programming problems gip with well determined duality gaps that is given an integer solution vector x our problem is to develop a set of integer linear inequalities ax b and an objective function c such that x lies within some known objective function distance of the optimal solution of the relaxed linear programming problem by well determined we mean that on completion an upper bound on the problem duality gap and an integer solution optimal or best known are available to the problem developer such a procedure can therefore be used to develop test problems to support the research effort in the area of general ip

exact numerical studies of hamiltonian maps iterating without roundoff error abstract for many important hamiltonian maps e g the standard map it is possible to construct related mappings that i carry a lattice into itself ii approach the original map as the lattice spacing is decreased iii can be iterated exactly using integer arithmetic and iv are hamiltonian themselves we compare these lattice maps to maps that use floating point arithmetic to evaluate the original map we discuss the problems associated with roundoff error and we argue that lattice maps are superior to floating point maps for the study of the long term behaviour of hamiltonian dynamical systems

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

a polyhedral study of the semi continuous knapsack problem we study the convex hull of the feasible set of the semi continuous knapsack problem in which the variables belong to the union of two intervals such structure arises in a wide variety of important problems e g blending and portfolio selection we show how strong inequalities valid for the semi continuous knapsack polyhedron can be derived and used in a branch and cut scheme for problems with semi continuous variables to demonstrate the effectiveness of these inequalities which we call collectively semi continuous cuts we present computational results on real instances of the unit commitment problem as well as on a number of randomly generated instances of linear programming with semi continuous variables

on the facets of the mixed integer knapsack polyhedron we study the mixed integer knapsack polyhedron that is the convex hull of the mixed integer set defined by an arbitrary linear inequality and the bounds on the variables we describe facet defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable these inequalities strengthen and or generalize known inequalities for several special cases we report computational results on using the inequalities as cutting planes for mixed integer programming

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

unstable periodic orbits and discretization cycles limit cycles that arise from discretizing the variable s of a nonlinear map are generally found to shadow individual unstable periodic orbits upos of the corresponding continuous map in a few cases the discretization cycles can only be explained with other mechanisms such as the near occurrence of an upo or crossover between two or more upos

effects of phase space discretization on the long time behavior of dynamical systems abstract we investigate the effects that roundoff errors have on discrete time dynamical systems computer generated orbits terminate in loops whose lengths scale with the machine precision in a universal way we present arguments to explain this fact

collapsing of chaos in one dimensional maps in their numerical investigation of the family of one dimensional maps f x d 1 2jxj where 2 diamond et al p diamond et al physica d 86 1999 559 571 have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at 1 a repelling fixed point this is a numerical artifact because the continuous maps are chaotic and almost every true trajectory can be shown to be dense in 1 1 the goal of this paper is to extend and resolve this obvious contradiction we model the numerical simulation with a randomly selected map while they used 27 bit precision in computing f we prove for our model that this numerical artifact persists for an arbitrary high numerical prevision the fraction of initial points eventually winding up at 1 remains bounded away from 0 for every numerical precision 2000 elsevier science b v all rights reserved

geometry and hamiltonian mechanics on discrete spaces numerical simulation is often crucial for analysing the behaviour of many complex systems which do not admit analytic solutions to this end one either converts a smooth model into a discrete in space and time model or models systems directly at a discrete level the goal of this paper is to provide a discrete analogue of differential geometry and to define on these discrete models a formal discrete hamiltonian structure in doing so we try to bring together various fundamental concepts from numerical analysis differential geometry algebraic geometry simplicial homology and classical hamiltonian mechanics for example the concept of a twisted derivation is borrowed from algebraic geometry for developing a discrete calculus the theory is applied to a nonlinear pendulum and we compare the dynamics obtained through a discrete modelling approach with the dynamics obtained via the usual discretization procedures also an example of an energy conserving algorithm on a simple harmonic oscillator is presented and its effect on the poisson structure is discussed

shadowing breakdown and large errors in dynamical simulations of physical systems simulations play a crucial role in the modern study of physical systems a major open question for long dynamical simulations of physical processes is the role of discretization and truncation errors in the outcome a general mechanism is described that can cause extremely small noise inputs to result in errors in simulation statistics that are several orders of magnitude larger a scaling law for the size of such errors in terms of the noise level and properties of the dynamics is given this result brings into question trajectory averages that are computed for systems with particular dynamical behaviors in particular systems that exhibit fluctuating lyapunov exponents or unstable dimension variability

incursive discretization system bifurcation and energy conservation incursive discretization of the classical harmonic oscillator leads to system bifurcation the resulting hyperincursive representation has two alternative distinct algorithms of ordered serial noncommuting instructions and admits solutions having a discretized classical total energy that is perfectly conserved and phase space trajectories that are fully stable at all time scales hyperincursive representations can be generated for any hamiltonian system

some inequalities about the kaplan meier estimator in this paper we consider the product limit estimator of the survival distribution function in the context of independent but nonidentically distributed censoring times an upper bound on the mean square increment of the stopped kaplan meier process is obtained also a representation is given for the ratio of the survival distribution function to the product limit estimator as the product of a bounded process and a martingale from this representation bounds on the mean square of the ratio and on the tail probability of the sup norm of the ratio are derived

a cutting plane approach to mixed 0 1 stochastic integer programs abstract we consider a mixed 0 1 integer programming problem with dual block angular structure arising in two stage stochastic programming a relaxation is proposed such that the problem is decomposed into subproblems each corresponding to the outcomes of the random variable the convex hull of feasible solutions for the relaxation is characterized using results from disjunctive programming and it is shown how lift and project cuts can be generated for one subproblem and made valid for different outcomes

noisy integer maps abstract the combined effects of discretization and noise in iterated maps are studied with simulations and a transfer matrix technique the resulting changes in a typical map s invariant distribution are illustrated

symplectic numerical integrators in constrained hamiltonian systems recent work reported in the literature suggests that for the long time integration of hamiltonian dynamical systems one should use methods that preserve the symplectic or canonical structure of the flow here we investigate the symplecticness of numerical integrators for constrained dynamics such as occur in molecular dynamics when bond lengths are made rigid in order to overcome stepsize limitations due to the highest frequencies this leads to a constrained hamiltonian system of smaller dimension previous work has shown that it is possible to have methods which are symplectic on the constraint manifold in phase space here it is shown that the very popular verlet method with shake type constraints is equivalent to the same method with rattle type constraints and that the latter is symplectic and time reversible this assumes that the iteration is carried to convergence we also demonstrate the global convergence of the verlet scheme in the presence of shake type and rattle type constraints we perform numerical experiments to compare these methods with the second order backward differentiation method commonly recommended for ordinary differential equations with constraints

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

machine iteration of a linear function local behaviour abstract we study analytically the behaviour of a discrete linear iteration near its fixed point in the case of truncation we also calculate the conditions needed for spurious multiple fixed points and for period two cycles we find that neither of these two errors can be reduced with better machine precision

using cutting planes in an interactive reference point approach for multiobjective integer linear programming problems we propose an interactive approach for multiple objective integer linear programming moilp problems that combines the use of the tchebycheff metric with cutting plane techniques at each interaction the method computes the nondominated solution for the moilp problem that is closest to a reference point according to the tchebycheff metric the information provided by the decision maker in each dialogue phase is used to adjust the next reference point through a sensitivity analysis stage cutting plane techniques enable the method to take advantage of computations performed at previous iterations to solve the next scalarizing integer program we address both theoretical issues and the computational implementation

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

limit cycles in a quadratic discrete iteration i study the truncated logistic equation as a map of d integers the number of limit cycles and the size of longest cycle are averaged exhaustively over many values of d for parameter values a beyond the first accumulation point fits of these quantities are compared with estimates obtained from random maps the results suggest some form of self organization i also present some analytical results on the existence and position of fixed points and a fast matrix method to enumerate limit cycles of arbitrary length for given a and d

physical tests for random numbers in simulations we propose three physical tests to measure correlations in random numbers used in monte carlo simulations the first test uses autocorrelation times of certain physical quantities when the ising model is simulated with the wolff algorithm the second test is based on random walks and the third on blocks of ital n successive numbers we apply the tests to show that recent errors in high precision ising simulations using generalized feedback shift register algorithms are due to short range correlations in random number sequences

chaos number theory and computers abstract a case study is presented in which the logistic map is shown to have anomalous precision dependent behavior it is also demonstrated that using greater precision does not necessarily lead to greater accuracy and in fact the contrary can be true though the presention deals only with this particular map it illustrates that the actual math package being used should be examined and cannot be treated as a black box

an efficient bound and stopped algorithm for integer linear programs on the objective function hyperplane this paper presents a new heuristic algorithm for general integer linear programming in which the objective function is varied as a parameter first of all since the feasible region of the linear programming relaxation problem is intercepted by the objective function hyperplane to form a convex polyhedra for some fixed integral value of the objective function the vertices on the polyhedron are determined and used to produce the value ranges of the variables by the expression of the linear convex combination of the vertices in the value ranges it can be identified whether a feasible solution to the integer program exists or not next a simple ingenious linear transformation of the original problem into new variables is made and a so called stopped search algorithm applied to finding the optimal solution or the fact that there is no feasible solution to the integer program on the objective function hyperplane comparing with some existing algorithms our algorithm is more easier in computation in that it must not get all the extreme points of the feasible domain of the relaxation problem and not solve otherwise ordinary linear programming subproblems and has a moderate memory requirement with the size of the problem the computational experience in the numerical examples shows that the algorithm presented here is efficient potential and hence is of great practical interest

fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems abstract branch and cut methods for integer programming problems solve a sequence of linear programming problems traditionally these linear programming relaxations have been solved using the simplex method the reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one if the solution to a relaxation is fractional additional constraints can be generated which cut off the solution to the relaxation but donot cut off any feasible integer points gomory cutting planes and other classes of cutting planes are generated from the final tableau in this paper we consider using an interior point method to solve the linear programming relaxations we show that it is still possible to generate gomory cuts and other cuts without having to recreate a tableau and we also show how variables can be fixed without using the optimal reduced costs the procedures we develop do not require that the current relaxation be solved to optimality this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators