an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

asymptotic properties of the positive equilibrium of a discrete survival model this paper is concerned with the discrete wazewska and lasota model x n 1 x email protected n pe n x n k where email protected 0 1 n email protected 0 and email protected 0 1 2 we show that every positive solution of is persistent and then we provide sufficient conditions for oscillation of all solutions about x finally we give sufficient conditions for x to be stable or attractive

global behavior of a higher order nonlinear difference equation abstract in this paper we consider a higher order difference equation of the form x n 1 f x n x n k n 1 2 3 under some certain assumptions we obtain that the length of its finite semicycle is less than or equal to k moreover we concluded that the equation is permanent under some special conditions and a sufficient condition for its global asymptotical stability is given when these results are applied to the difference equation x n 1 x n k x n n 1 2 3 some sufficient and necessary conditions of global asymptotical stability of the equation are obtained we should state out that the main theorem in j math anal appl 233 1999 790 798 is included in one of our results

improved railway wheelset track interaction model in the high frequency domain as it is well known there are various phenomena related to railway train track interaction some of them caused by the high frequency dynamics of the system such as rolling noise when the vehicle runs over the track as well as squeal noise and short pitch rail corrugation for curved tracks due to these phenomena and some others unsolved so far a large effort has been made over the last 40 years in order to define suitable models to study the train track interaction the introduction of flexibility in wheelset and rail models was required to have a more realistic representation of the wheel rail interaction effects at high frequencies in recently published train track interaction models the rails are modelled by means of timoshenko beam elements valid up to 1 5 khz for lateral rail vibration and up to 2 khz for vertical vibration this confines the frequency range of validity for the complete train track model to 1 5 khz with the purpose of extending the range of validity above 1 5 khz a 3d track model based on the moving element method mem is developed in this paper to replace the timoshenko beam considered in earlier studies adopting cyclic boundary conditions and eulerian coordinates the mem approach considers a mobile finite element fe mesh which moves with the vehicle so the mass of the rail flows with the vehicle speed but in the opposite direction through the mesh therefore the mem permits to fix the contact area in the middle of a finitely long track and to refine the mesh only around the contact area where the forces and displacements will be more significant additionally a modal approach is adopted in order to reduce the number of degrees of freedom of the rail model both strategies lower substantially the computational cost simulation results are presented and discussed for different excitation sources including random rail roughness and singularities such as wheel flats all the simulation cases are carried out for a timoshenko beam and a 3d mem track model in order to point out the differences in the contact forces above the range of validity of the timoshenko beam 3d moving element method mem to model the track is proposed the model considers an eulerian coordinate system attached to the moving vehicle the resulting formulation permits to reduce the computational cost compared to the fe models commonly used the proposed 3d mem track model is suitable to describe the high frequency dynamics

a polynomial time interior point method for conic optimization with inexact barrier evaluations we consider a primal dual short step interior point method for conic convex optimization problems for which exact evaluation of the gradient and hessian of the primal and dual barrier functions is either impossible or prohibitively expensive as our main contribution we show that if approximate gradients and hessians of the primal barrier function can be computed and the relative errors in such quantities are not too large then the method has polynomial worst case iteration complexity in particular polynomial iteration complexity ensues when the gradient and hessian are evaluated exactly in addition the algorithm requires no evaluation or even approximate evaluation of quantities related to the barrier function for the dual cone even for problems in which the underlying cone is not self dual

a numerical method for moving load on continuum abstract in this paper a recently developed method called the moving element method is adopted for the dynamic analysis of half space continuum under moving load plane strain condition is assumed and the continuum is discretised into moving elements these moving elements are not physical elements fixed to the continuum but are conceptual elements that flow with the moving load through the continuum the method eliminates the need of keeping track the location of moving load relative to the element mesh numerical examples involving different load types such as strip load and concentrated load moving on half space and layers on half space are presented the numerical results are verified against solutions obtained by alternative methods

relaxed variants of karmarkar s algorithm for linear programs with unknown optimal objective value variants of karmarkar s algorithm are given for solving linear programs with unknown optimal objective valuez these new methods combine the approach of goldfarb and mehrotra for relaxing the requirement that certain projections be computed exactly with the approach of todd and burrell for generating an improving sequence of lower bounds forz using dual feasible solutions these methods retain the polynomial time complexity of karmarkar s algorithm

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

a relaxed revision of karmarkar s method a relaxed version of karmarkar s method is developed this method is proved to have the same polynomial time complexity as karmarkar s method and its efficient implementation using inexact projections is discussed computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

global stability and oscillation in nonlinear difference equations of population dynamics in this paper we study the qualitative behavior of solutions of the discrete population modelx n x n 1 px n qx nr x n k m n 0 1 2 where email protected 0 1 q email protected 0 p

a qmr based interior point algorithm for solving linear programs a new approach for the implementation of interior point methods for solving linear programs is proposed its main feature is the iterative solution of the symmetric but highly indefinite 2 2 block systems of linear equations that arise within the interior point algorithm these linear systems are solved by a symmetric variant of the quasi minimal residual qmr algorithm which is an iterative solver for general linear systems the symmetric qmr algorithm can be combined with indefinite preconditioners which is crucial for the efficient solution of highly indefinite linear systems yet it still fully exploits the symmetry of the linear systems to be solved to support the use of the symmetric qmr iteration a novel stable reduction of the original unsymmetric 3 3 block systems to symmetric 2 2 block systems is introduced and a measure for a low relative accuracy for the solution of these linear systems within the interior point algorithm is proposed some indefinite preconditioners are discussed finally we report results of a few preliminary numerical experiments to illustrate the features of the new approach

global attractivity in a nonlinear second order difference equation in this paper we obtain a global attractivity result for the positive equilibrium of a nonlinear second order difference equation of the form xn 1 f xn xn 1 n 0 1 the result applies to the difference equation xn 1 a bxn a n 1 n 0 1 where a b a 0 1996 john wiley sons inc

correlation clustering with constrained cluster sizes and extended weights bounds we consider the problem of correlation clustering on graphs with constraints on both the cluster sizes and the positive and negative weights of edges our contributions are twofold first we introduce the problem of correlation clustering with bounded cluster sizes second we extend the regime of weight values for which the clustering may be performed with constant approximation guarantees in polynomial time and apply the results to the bounded cluster size problem

computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming an example of the quadratic assignment problem qap is the facility location problem in which n facilities are assigned at minimum cost to n sites between each pair of facilities there is a given amount of flow contributing a cost equal to the product of the flow and the distance between sites to which the facilities are assigned proving optimality of qaps has been limited to instances having fewer than 20 facilities largely because known lower bounds are weak we compute lower bounds for a wide range of qaps using a linear programming based lower bound studied by z drezner 1995 on the majority of qaps tested a new best known lower bound is computed on 87 of the instances we produced the best known lower bound on several instances including some having more the 20 facilities the lower bound is tight the linear programs which can be large even for small qaps are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions attempts to solve these instances using the cplex primal simplex algorithm as well as the cplex barrier primal dual interior point method were successful only for the smallest instances

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 acyclic subgraph polytope the acyclic subgraph problem can be formulated as follows given a digraph with arc weights find a set of arcs containing no directed cycle and having maximum total weight we investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope we also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time this implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs this generalizes a result of lucchesi for planar digraphs

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

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

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems

a variable metric variant of the karmarkar algorithm for linear programming the most time consuming part of the karmarkar algorithm for linear programming is the projection of a vector onto the nullspace of a matrix that changes at each iteration we present a variant of the karmarkar algorithm that uses standard variable metric techniques in an innovative way to approximate this projection in limited tests this modification greatly reduces the number of matrix factorizations needed for the solution of linear programming problems