optimal solution of vehicle routing problems using minimum k trees we consider the problem of optimally scheduling a fleet of k vehicles to make deliveries to n customers subject to vehicle capacity constraints given a graph with n 1 nodes a k tree is defined to be a set of n k edges that span the graph we show that the vehicle routing problem can be modeled as the problem of finding a minimum cost k tree with two k edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once the side constraints are dualized to obtain a lagrangian problem that provides lower bounds in a branch and bound algorithm this algorithm has produced proven optimal solutions for a number of difficult problems including a well known problem with 100 customers and several real problems with 25 71 customers

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems abstract we propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems these formulations are obtained by lifting using the reformulation linearization technique a novel polynomial size ordered path based formulation we show that the strongest proposed formulation is equivalent to the strongest multi commodity flow formulation presented so far in addition we propose polynomial size valid inequalities that aim at further tightening the proposed formulations also the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs we show that the new derived model is not comparable with a state of the art polynomial length formulation we present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities

optimal routing under capacity and distance restrictions this paper describes an integer linear programming algorithm for vehicle routing problems involving capacity and distance constraints the method uses constraint relaxation and a new class of subtour elimination constraints two versions of the algorithm are presented depending upon the nature of the distance matrix exact solutions are obtained for problems involving up to sixty cities

an exact algorithm for the capacitated vehicle routing problem based on a two commodity network flow formulation the capacitated vehicle routing problem cvrp is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints in this paper we describe a new integer programming formulation for the cvrp based on a two commodity network flow approach we present a lower bound derived from the linear programming lp relaxation of the new formulation which is improved by adding valid inequalities in a cutting plane fashion moreover we present a comparison between the new lower bound and lower bounds derived from the lp relaxations of different cvrp formulations proposed in the literature a new branch and cut algorithm for the optimal solution of the cvrp is described computational results are reported for a set of test problems derived from the literature and for new randomly generated problems

polyhedral results for a vehicle routing problem abstract the vehicle routing problem is a well known and hard combinatorial problem whose polyhedral structure has deserved little attention in this paper we consider the particular case in which all the demands are equal since in the general case the associated polytope may be empty from a known formulation of the problem we obtain the dimension of the corresponding polytope and we study the facetial properties of every inequality in it

an exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts this paper presents a new exact algorithm for the capacitated vehicle routing problem cvrp based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities the exact algorithm uses a bounding procedure that finds a near optimal dual solution of the lp relaxation of the resulting mathematical formulation by combining three dual ascent heuristics the first dual heuristic is based on the q route relaxation of the set partitioning formulation of the cvrp the second one combines lagrangean relaxation pricing and cut generation the third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique the final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved the resulting problem is solved by an integer programming solver computational results over the main instances from the literature show the effectiveness of the proposed algorithm

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

computational experience with stable set relaxations we investigate relaxations for the maximum stable set problem based on the lovasz number vartheta g as an initial upper bound we strengthen this relaxation by adding two classes of cutting planes odd circuit and triangle inequalities we present computational results using this tighter model on many classes of graphs

the symmetric traveling salesman polytope and its graphical relaxation composition of valid inequalities the graphical relaxation of the traveling salesman problem is the relaxation obtained by requiring that the salesman visit each city at least once instead of exactly once this relaxation has already led to a better understanding of the traveling salesman polytope in cornuejols fonlupt and naddef 1985 we show here how one can compose facet inducing inequalities for the graphical traveling salesman polyhedron and obtain other facet inducing inequalities this leads to new valid inequalities for the symmetric traveling salesman polytope this paper is the first of a series of three papers on the symmetric traveling salesman polytope the next one studies the strong relationship between that polytope and its graphical relaxation and the last one applies all the theoretical developments of the two first papers to prove some new facet inducing results

stronger k tree relaxations for the vehicle routing problem abstract a lagrangian based exact solution algorithm for the vehicle routing problem vrp defined on an undirected graph is introduced in this paper lower bounds are obtained by allowing exponentially many inequalities as candidates to lagrangian dualization three different families of strong valid inequalities each one with exponentially many elements are used within vrp formulations for each of them separation procedures are proposed for points that define incidence vectors of k trees violated inequalities identified in this way are then dualized in a relax and cut framework upper bounds are generated through a lagrangian clarke and wright heuristic a variable fixation test based on approximating linear programming reduced costs is also implemented computational results are presented for the proposed algorithm

connections between semidefinite relaxations of the max cut and stable set problems we describe links between a recently introduced semidefinite relaxation for the max cut problem and the well known semidefinite relaxation for the stable set problem underlying the lovasz s theta function it turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra we also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

bilevel programming and the separation problem in recent years branch and cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs milps methods for automatically generating inequalities valid for the convex hull of solutions to such milps are a critical element of branch and cut this paper examines the nature of the so called separation problem which is that of generating a valid inequality violated by a given real vector usually arising as the solution to a relaxation of the original problem we show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program in some cases this bilevel program can be easily reformulated as a simple single level mathematical program yielding a standard mathematical programming formulation for the separation problem in other cases no such polynomial size single level reformulation exists unless the polynomial hierarchy collapses to its first level an event considered extremely unlikely in computational complexity theory we illustrate our insights by considering the separation problem for two well known classes of valid inequalities

a collocated grid finite volume method for aeroacoustic computations of low speed flows a numerical algorithm for simulation of acoustic noise generation based on collocated grids is described the approach that was originally developed using a viscous inviscid decomposition technique involves two steps comprising a viscous incompressible flow part and an inviscid or viscous acoustic part on collocated grids the inviscid solution is found to be mesh dependent due to unavoidable extrapolations of the acoustic pressure and density at walls differing from the case on staggered grids where no extrapolation is needed the situation is most pronounced when a sharp body is considered a viscous acoustic algorithm is proposed to overcome the difficulty numerical computations of flows past a circular cylinder and a naca 0015 airfoil show that a viscous viscous coupling is more natural and gives excellent results as compared to those obtained in previous computations based on viscous inviscid coupling on staggered grids the model is applied to the problem of an airfoil exposed to a gust and results are compared to the numerical results of lockard and morris aiaa j 36 6 1998 907

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

separating capacity constraints in the cvrp using tabu search abstract branch and cut methods have shown to be very successful in the resolution of some hard combinatorial optimization problems the success has been remarkable for the symmetric traveling salesman problem tsp the crucial part in the method is the cutting plane algorithm the algorithm that looks for valid inequalities that cut off the current nonfeasible linear program lp solution in turn this part relies on a good knowledge of the corresponding polyhedron and our ability to design algorithms that can identify violated valid inequalities this paper deals with the separation of the capacity constraints for the capacitated vehicle routing problem cvrp three algorithms are presented a constructive algorithm a randomized greedy algorithm and a very simple tabu search procedure as far as we know this is the first time a metaheuristic is used in a separation procedure the aim of this paper is to present this application no advanced tabu technique is used we report computational results with these heuristics on difficult instances taken from the literature as well as on some randomly generated instances these algorithms were used in a branch and cut procedure that successfully solved to optimality large cvrp instances

a numerical method for solving the equations of compressible viscous flow although much progress has already been made in solving problems in aerodynamic design many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely this paper describes one such development a new method for solving these equations has been devised that 1 is second order accurate in space and time 2 is unconditionally stable 3 preserves conservation form 4 requires no block or scalar tridiagonal inversions 5 is simple and straightforward to program estimated 10 modification for the update of many existing programs 6 is more efficient than present methods and 7 should easily adapt to current and future computer architectures computational results for laminar and turbulent flows at reynolds numbers from 3 x 10 exp 5 to 3 x 10 exp 7 and at cfl numbers as high as 10 exp 3 are compared with theory and experiment

a lower bound for the split delivery vehicle routing problem in this paper we consider the split delivery vehicle routing problem sdvrp a relaxation of the known capacitated vehicle routing problem cvrp in which the demand of any client can be serviced by more than one vehicle we define a feasible solution of this problem and we show that the convex hull of the associated incidence vectors is a polyhedron psdvrp whose dimension depends on whether a vehicle visiting a client must service or not at least one unit of the client demand from a partial and linear description ofpsdvrp and a new family of valid inequalities we develop a lower bound whose quality is exhibited in the computational results provided which include the optimal resolution of some known instances one of them with 50 clients this instance is as far as we know the biggest one solved so far

a result on projection for the vehicle routing ptoblem abstract in this paper we present a result on projection for the vehicle routing problem vrp the vrp is closely related to delivery type problems and appears in a large number of practical situations concerning the distribution of commodities the present work focuses on a commodity flow formulation presented by gavish and graves this formulation includes two sets of variables and hence it also must include coupling constraints between the two sets of variables these coupling constraints can be defined in several ways the main result of this work establishes that when the strongest form of the coupling constraints is used in the flow formulation the equivalent formulation using only the x ij variables satisfies the so called multistar constraints which for certain parameters induce facets of the non directed vrp polytope using an idea taken from gouveia we show how to derive a more compact representation in the number of constraints of the multistar constraints some consequences of our projection result are also discussed

an algebraic subgrid scale finite element method for the convected helmholtz equation in two dimensions with applications in aeroacoustics abstract an algebraic subgrid scale finite element method formally equivalent to the galerkin least squares method is presented to improve the accuracy of the galerkin finite element solution to the two dimensional convected helmholtz equation a stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method an appropriate value for this parameter has been obtained by means of a dispersion analysis as an application we have considered the case of aerodynamic sound radiated by incompressible flow past a two dimensional cylinder following lighthill s acoustic analogy we have used the time fourier transform of the double divergence of the reynolds stress tensor as a source term for the helmholtz and convected helmholtz equations and showed the benefits of using the subgrid scale stabilization

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

development of computational aeroacoustics equations for subsonic flows using a mach number expansion approach computational aeroacoustics equations are developed using a janzen rayleigh expansion of the compressible flow equations separate expansions are applied to an inner region characterized to lowest order by an incompressible flow field and an outer region characterized by propagating acoustic waves several perturbation equation sets are developed in the inner and outer regions by truncating the expanded equations using different orders in the perturbation variable where 2 is proportional to the square of the mach number characterizing the flow composite equation sets are constructed by matching the equations governing the inner and outer regions the highest order perturbation continuity and momentum equations include an infinite series in 2 and are shown to be identical to the equations used in the expansion about incompressible flow approach as such the perturbation analysis is used to interpret the physical meaning of the perturbation variables and to highlight the assumptions inherent in this approach differences between numerical solutions obtained with the composite equation sets are evaluated for two unsteady flow problems the lowest order perturbation equation set is shown to yield adequate acoustic predictions for low mach number flows this equation set is considerably simpler to implement into a numerical solver and reduces the required cpu time relative to the highest order equation set

some thoughts on combinatorial optimisation abstract a group of young researchers from the esi x summer school hec jouy en josas 1994 give their personal views on the current status of and prospects for combinatorial optimisation several issues are considered and discussed with emphasis on a selected number of techniques heuristics and polyhedral approaches and problems knapsack quadratic 0 1 programming machine scheduling routing and network design

a heuristic algorithm for the asymmetric capacitated vehicle routing problem abstract we consider the asymmetric capacitated vehicle routing problem acvrp a particular case of the standard asymmetric vehicle routing problem arising when only the vehicle capacity constraints are imposed acvrp is known to be np hard and finds practical applications e g in distribution and scheduling in this paper we describe the extension to acvrp of the two well known clarke wright and fisher jaikumar heuristic algorithms we also propose a new heuristic algorithm for acvrp that starting with an initial infeasible solution determines the final set of vehicle routes through an insertion procedure as well as intea route and inter route arc exchanges the initial infeasible solution is obtained by using the additive bounding procedures for acvrp described by fischetti toth and vigo in 1992 extensive computational results on several classes of randomly generated test problems involving up to 300 customers and on some real instances of distribution problems in urban areas are presented the results obtained show that the proposed approach favourably compares with previous algorithms from the literature

numerical study of sound generation due to a spinning vortex pair a numerical approach for the calculation of an acoustic field is applied to the case of a spinning vortex pair to investigate the sound generation by quadrupole sources in unsteady vortical flows based on the unsteady hydrodynamic information from the known incompressible flowfield the perturbed compressible acoustic terms derived from the euler equations are calculated nonreflecting boundary conditions are developed to obtain highly stable solutions calculated results are compared with analytical solutions obtained by the method of matched asymptotic expansions the possibility of predicting the effect of convective mean flow is also tested it is concluded that the sound generated by the quadrupole sources of unsteady vortical flows without a sound generating body or surface can be calculated by using the source terms of hydrodynamic flow fluctuations

polyhedral study of the capacitated vehicle routing problem the capacitated vehicle routing problem cvrp considered in this paper occurs when goods must be delivered from a central depot to clients with known demands usingk vehicles of fixed capacity each client must be assigned to exactly one of the vehicles the set of clients assigned to each vehicle must satisfy the capacity constraint the goal is to minimize the total distance traveled when the capacity of the vehicles is large enough this problem reduces to the famous traveling salesman problem tsp a variant of the problem in which each client is visited by at least one vehicle called the graphical vehicle routing problem gvrp is also considered in this paper and used as a relaxation of cvrp our approach for cvrp and gvrp is to extend the polyhedral results known for tsp for example the subtour elimination constraints can be generalized to facets of both cvrp and gvrp interesting classes of facets arise as a generalization of the comb inequalities depending on whether the depot is in a handle a tooth both or neither we report on the optimal solution of two problem instances by a cutting plane algorithm that only uses inequalities from the above classes

regularization methods for semidefinite programming we introduce a new class of algorithms for solving linear semidefinite programming sdp problems our approach is based on classical tools from convex optimization such as quadratic regularization and augmented lagrangian techniques we study the theoretical properties and we show that practical implementations behave very well on some instances of sdp having a large number of constraints we also show that the boundary point method from povh rendl and wiegele computing 78 2006 pp 277 286 is an instance of this class

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

a tabu search heuristic for the vehicle routing problem the purpose of this paper is to describe taburoute a new tabu search heuristic for the vehicle routing problem with capacity and route length restrictions the algorithm considers a sequence of adjacent solutions obtained by repeatedly removing a vertex from its current route and reinserting it into another route this is done by means of a generalized insertion procedure previously developed by the authors during the course of the algorithm infeasible solutions are allowed numerical tests on a set of benchmark problems indicate that tabu search out performs the best existing heuristics and taburoute often produces the best known solutions

a branch first cut second approach for locomotive assignment the problem of assigning locomotives to trains consists of selecting the types and number of engines that minimize the fixed and operational locomotive costs resulting from providing sufficient power to pull trains on fixed schedules the force required to pull a train is often expressed in terms of horsepower and tonnage requirements rather than in terms of number of engines this complicates the solution process of the integer programming formulation and usually creates a large integrality gap furthermore the solution of the linearly relaxed problem is strongly fractional to obtain integer solutions we propose a novel branch and cut approach the core of the method consists of branching decisions that define on one branch the projection of the problem on a low dimensional subspace there the facets of the polyhedron describing a restricted constraint set can be easily derived we call this approach branch first cut second we first derive facets when at most two types of engines are used we then extend the branching rule to cases involving additional locomotive types we have conducted computational experiments using actual data from the canadian national railway company simulated test problems involving two or more locomotive types were solved over 1 2 and 3 day rolling horizons the cuts were successful in reducing the average integrality gap by 52 for the two type case and by 34 when more than 25 types were used furthermore the branch first cut second approach was instrumental in improving the best known solution for an almost 2 000 leg weekly problem involving 26 locomotive types it reduced the number of locomotives by 11 or 1 1 at an equivalent savings of 3 000 000 per unit additional tests on different weekly data produced almost identical results

a new branch and cut algorithm for the capacitated vehicle routing problem we present a new branch and cut algorithm for the capacitated vehicle routing problem cvrp the algorithm uses a variety of cutting planes including capacity framed capacity generalized capacity strengthened comb multistar partial multistar extended hypotour inequalities and classical gomory mixed integer cuts for each of these classes of inequalities we describe our separation algorithms in detail also we describe the other important ingredients of our branch and cut algorithm such as the branching rules the node selection strategy and the cut pool management computational results for a large number of instances show that the new algorithm is competitive in particular we solve three instances b n50 k8 b n66 k9 and b n78 k10 of augerat to optimality for the first time

multistars partial multistars and the capacitated vehicle routing problem in an unpublished paper araque hall and magnanti considered polyhedra associated with the capacitated vehicle routing problem cvrp in the special case of unit demands among the valid and facet inducing inequalities presented in that paper were the so called multistar and partial multistar inequalities each of which came in several versions some related inequalities for the case of general demands have appeared subsequently and the result is a rather bewildering array of apparently different classes of inequalities the main goal of the present paper is to present two relatively simple procedures that can be used to show the validity of all known and some new multistar and partial multistar inequalities in both the unit and general demand cases the procedures provide a unifying explanation of the inequalities and perhaps more importantly ideas that can be exploited in a cutting plane algorithm for the cvrp computational results show that the new inequalities can be useful as cutting planes for certain cvrp instances

a tabu search heuristic for the vehicle routing problem with stochastic demands and customers this paper considers a version of the stochastic vehicle routing problem where customers are present at locations with some probabilities and have random demands a tabu search heuristic is developed for this problem comparisons with known optimal solutions on problems whose sizes vary from 6 to 46 customers indicate that the heuristic produces an optimal solution in 89 45 of cases with an average deviation of 0 38 from optimality

robust branch and cut and price for the capacitated vehicle routing problem the best exact algorithms for the capacitated vehicle routing problem cvrp have been based on either branch and cut or lagrangean relaxation column generation this paper presents an algorithm that combines both approaches it works over the intersection of two polytopes one associated with a traditional lagrangean relaxation over q routes the other defined by bound degree and capacity constraints this is equivalent to a linear program with exponentially many variables and constraints that can lead to lower bounds that are superior to those given by previous methods the resulting branch and cut and price algorithm can solve to optimality all instances from the literature with up to 135 vertices this more than doubles the size of the instances that can be consistently solved

reactive linearized equations of perturbed compressible variables for low mach number variable density flows a hydrodynamic acoustic splitting approach is proposed to study noise emitted from reactive variable density flows a simulation using the variable density low mach number equations provides a solution of the hydrodynamic motions of the flow base flow and a set of equations for perturbed variables is additionally solved to capture the acoustic motions a rigorous derivation of these equations for the assumed base flow is given which compared to its non reacting counterpart includes additional terms related to variable density flows two different test cases are presented first the kirchhoff vortex is simulated to highlight instability issues related to constant density flows second an academic test case for variable density flows is proposed in the form of a reacting dipole which is used to underline the stability of the proposed perturbed equations in such a scenario various intermediate forms of the derived perturbation equations are juxtaposed and analyzed with respect to their stability for these two test cases and assumptions made in their derivation are numerically justified

on the capacitated vehicle routing problem we consider the vehicle routing problem in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost this difficult combinatorial problem contains both the bin packing problem and the traveling salesman problem tsp as special cases and conceptually lies at the intersection of these two well studied problems the capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures we describe a decomposition based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the tsp efficiently specifically when standard procedures fail to separate a candidate point we attempt to decompose it into a convex combination of tsp tours if successful the tours present in this decomposition are examined for violated capacity constraints if not the farkas theorem provides a hyperplane separating the point from the tsp polytope we present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models computational results are given for an implementation within the parallel branch cut and price framework symphony

an application of the lovasz schrijver m k k operator to the stable set problem although the lift and project operators of lovasz and schrijver have been the subject of intense study their m k k operator has received little attention we consider an application of this operator to the stable set problem we begin with an initial linear programming lp relaxation consisting of clique and non negativity inequalities and then apply the operator to obtain a stronger extended lp relaxation we discuss theoretical properties of the resulting relaxation describe the issues that must be overcome to obtain an effective practical implementation and give extensive computational results remarkably the upper bounds obtained are sometimes stronger than those obtained with semidefinite programming techniques