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

a branch and cut approach to the cardinality constrained circuit problem the cardinality constrained circuit problem cccp is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges the cccp is np hard we present classes of facet inducing inequalities for the convex hull of feasible circuits and a branch and cut solution approach using these inequalities

the traveling salesman problem an overview of exact and approximate algorithms in this paper some of the main known algorithms for the traveling salesman problem are surveyed the paper is organized as follows 1 definition 2 applications 3 complexity analysis 4 exact algorithms 5 heuristic algorithms 6 conclusion

solving matching problems with linear programming in this paper we describe an implementation of a cutting plane algorithm for the perfect matching problem which is based on the simplex method the algorithm has the following features it works on very sparse subgraphs ofkn which are determined heuristically global optimality is checked using the reduced cost criterion cutting plane recognition is usually accomplished by heuristics only if these fail the padberg rao procedure is invoked to guarantee finite convergence

a lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets given any family of valid inequalities for the asymmetric traveling salesman polytopep g defined on the complete digraphg we show that all members of are facet defining if the primitive members of usually a small subclass are based on this result we then introduce a general procedure for identifying new classes of facet inducing inequalities forp g by lifting inequalities that are facet inducing forp g whereg is some induced subgraph ofg unlike traditional lifting where the lifted coefficients are calculated one by one and their value depends on the lifting sequence our lifting procedure replaces nodes ofg with cliques ofg and uses closed form expressions for calculating the coefficients of the new arcs which are sequence independent we also introduce a new class of facet inducing inequalities the class of sd source destination inequalities which subsumes as special cases most known families of facet defining inequalities

a fast optimization method based on a hierarchical strategy for the travelling salesman problem abstract an algorithm based on a hierarchical strategy is proposed for solving the travelling salesman problem the tour patterns have been analyzed and compared with those generated from different stages of simulated annealing a set of problems with known solutions has been used to evaluate the new algorithm the hierarchical algorithm is shown to be very fast it finds a path length similar to that obtained using simulated annealing algorithm the path length is quite far from the global minimum but compares favorably to that obtained using simulated annealing

the merchant subtour problem we consider the problem of a travelling merchant who makes money by buying commodities where they are cheap and selling them in other places where he can make a profit the merchant ships commodities of his own choice in a van of fixed capacity given the prices of all the commodities in all of the places and the cost of driving from one place to another the problem the merchant faces each day is to select a subset of the cities that he can visit in a day and to determine the order in which the cities are visited such that the total profit is maximised we call this problem the merchant subtour problem the msp models the pricing problem of a rather complex pickup and delivery problem that was given to us by the dutch logistics company van gend loos we show that a special case of the msp has a totally unimodular constraint matrix this knowledge enables us to develop a tabu search algorithm for finding good feasible solutions to the msp and a branch and price and cut algorithm for solving the msp to optimality the relaxations solved in each node of the branch and bound tree are strengthened by lifted knapsack inequalities lifted cycle inequalities and mod k cuts we present computational results on data sets derived from our main instance of the van gend loos pickup and delivery problem

analysis of a finite pml approximation to the three dimensional elastic wave scattering problem we consider the application of a perfectly matched layer pml technique to approximate solutions to the elastic wave scattering problem in the frequency domain the pml is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain the complement of the scatterer the rapid decay of the pml solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation for the truncated problem we prove existence and uniqueness of the solutions to the infinite domain and truncated domain pml equations provided that the truncated domain is sufficiently large we also show exponential convergence of the solution of the truncated pml problem to the solution of the original scattering problem in the region of interest we then analyze a galerkin numerical approximation to the truncated pml problem and prove that it is well posed provided that the pml damping parameter and mesh size are small enough finally computational results illustrating the efficiency of the finite element pml approximation are presented

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

a new approximation technique for div curl systems in this paper we describe an approximation technique for div curl systems based in l where is a domain in 3 we formulate this problem as a general variational problem with different test and trial spaces the analysis requires the verification of an appropriate inf sup condition this results is a very weak formulation where the solution space is l and the data reside in various negative norm spaces subsequently we consider finite element approximations based on this weak formulation the main approach of this paper involves the development of stable pairs of discrete test and trial spaces with this approach we enlarge the test space so that the discrete inf sup condition holds and use a negative norm least squares formulation to reduce to a uniquely solvable linear system this leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low sobolev regularity e g h with 0 s 1 2 the resulting algebraic equations are symmetric positive definite and well conditioned a second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned some numerical results are also presented

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

easily computable facets of the knapsack polytope it is known that facets and valid inequalities for the knapsack polytope p can be obtained by lifting a simple inequality derived from each minimal cover we study the computational complexity of such lifting in particular we show that the task of computing a lifted facet can be accomplished in o ns where s n is the cardinality of the minimal cover also for a lifted inequality with integer coefficients we show that the dual tasks of recognizing whether the inequality is valid for p or is a facet of p can be done within the same time bound

on the separation of maximally violated mod k cuts separation is of fundamental importance in cutting plane based techniques for integer linear programming ilp in recent decades a considerable research effort has been devoted to the definition of effective separation procedures for families of well structured cuts in this paper we address the separation of chvatal rank 1 inequalities in the context of general ilp s of the form min c t x ax b x integer where a is an m n integer matrix and b an m dimensional integer vector in particular for any given integer k we study mod k cuts of the form t ax t b for any 0 1 k k 1 k m such that t a is integer following the line of research recently proposed for mod 2 cuts by applegate bixby chvatal and cook 1 and fleischer and tardos 19 we restrict to maximally violated cuts i e to inequalities which are violated by k 1 k by the given fractional point we show that for any given k such a separation requires o mn min m n time applications to both the symmetric and asymmetric tsp are discussed in particular for any given k we propose an o v 2 e time exact separation algorithm for mod k cuts which are maximally violated by a given fractional symmetric or asymmetric tsp solution with support graph g v e this implies that we can identify a maximally violated cut for the symmetric tsp whenever a maximally violated extended comb inequality exists finally facet defining mod k cuts for the symmetric and asymmetric tsp are studied

polynomial time separation of a superclass of simple comb inequalities the comb inequalities are a well known class of facet inducing inequalities for the traveling salesman problem defined in terms of certain vertex sets called the handle and the teeth we say that a comb inequality is simple if the following holds for each tooth either the intersection of the tooth with the handle has cardinality one or the part of the tooth outside the handle has cardinality one or both the simple comb inequalities generalize the classical 2 matching inequalities of edmonds edmonds j 1965 maximum matching and a polyhedron with 0 1 vertices j res nat bur standards69b 125 130 and also the so called chvatal comb inequalities in 1982 padberg and rao padberg m w m r rao 1982 odd minimum cut sets and b matchings math oper res 7 67 80 gave a polynomial time separation algorithm for the 2 matching inequalities i e an algorithm for testing if a given fractional solution to an lp relaxation violates a 2 matching inequality we extend this significantly by giving a polynomial time separation algorithm for a class of valid inequalities which includes all simple comb inequalities

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

the crown inequalities for the symmetric traveling salesman polytope we define a new family of valid inequalities for the symmetric traveling salesman polytope stspn that is the convex hull of the incidence vectors of all hamiltonian cycles of a complete graph with n nodes the smallest value of n for which the inequalities are defined is 8 we call these inequalities crown inequalities and we prove that they are facet inducing for stspn for n 8 we describe some extensions of the crown inequalities and we prove that they induce facets of stspn as well finally we discuss some issues on the possible applications of these inequalities in a polyhedral cutting plane algorithm

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

an optimality cut for mixed integer linear programs abstract we derive the penalty cut a simple optimality cut of general applicability in pure or mixed linear programs this cut is tested on a number of examples and comparisons with the classical gomory cut are provided

the pickup and delivery problem faces and branch and cut algorithm abstract this paper formulates the pickup and delivery problem also known as the dial a ride problem as an integer program its polyhedral structure is explored and four classes of valid inequalities developed the results of a branch and cut algorithm based on these constraints are presented

a cutting plane algorithm for the general routing problem the general routing problem grp is the problem of finding a minimum cost route for a single vehicle subject to the condition that the vehicle visits certain vertices and edges of a network it contains the rural postman problem chinese postman problem and graphical travelling salesman problem as special cases we describe a cutting plane algorithm for the grp based on facet inducing inequalities and show that it is capable of providing very strong lower bounds and in most cases optimal solutions

on the domino parity inequalities for the stsp one method which has been used very successfully for finding optimal and provably good solutions for large instances of the symmetric travelling salesman problem stsp is the branch and cut method this method requires knowledge of classes of useful valid inequalities for the polytope associated with the stsp as well as efficient separation routines for these classes of inequalities recently a new class of valid inequalities called the domino parity inequalites were introduced for the stsp an efficient separation routine is known for these constraints if certain conditions are satisfied by the point to be separated this separation routine has never been implemented or tested we present several performance enhancements for this separation routine and discuss our implementation of this improved algorithm we test our implementation and provide results which we believe demonstrate the practical usefulness of these constraints and the separation routine for the stsp within a branch and cut framework

the precedence constrained asymmetric traveling salesman polytope many applications of the traveling salesman problem require the introduction of additional constraints one of the most frequently occurring classes of such constraints are those requiring that certain cities be visited before others precedence constraints in this paper we study the precedence constrained asymmetric traveling salesman pcats polytope i e the convex hull of incidence vectors of tours in a precedence constrained directed graph we derive several families of valid inequalities and give polynomial time separation algorithms for important subfamilies we then establish the dimension of the pcats polytope and show that under reasonable assumptions the two main classes of inequalities derived are facet inducing

separating clique trees and bipartition inequalities having a fixed number of handles and teeth in polynomial time many important cutting planes have been discovered for the traveling salesman problem among them are the clique tree inequalities and the more general bipartition inequalities little is known in the way of exact algorithms for separating these inequalities in polynomial time in the size of the fractional point x which is being separated an algorithm is presented here that separates bipartition inequalities of a fixed number of handles and teeth which includes clique tree inequalities of a fixed number of handles and teeth in polynomial time

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

on the separation of split cuts and related inequalities the split cuts of cook kannan and schrijver are general purpose valid inequalities for integer programming which include a variety of other well known cuts as special cases to detect violated split cuts one has to solve the associated separation problem the complexity of split cut separation was recently cited as an open problem by cornuejols li cl01 in this paper we settle this question by proving strong completeness of separation for split cuts as a by product we also show completeness of separation for several other classes of inequalities including the mir inequalities of nemhauser and wolsey and some new inequalities which we call balanced split cuts and binary split cuts we also strengthen completeness results of caprara fischetti cf96 for cuts and eisenbrand e99 for chvatal gomory cuts to compensate for this bleak picture we also give a positive result for the symmetric travelling salesman problem we show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

a polyhedral approach to the asymmetric traveling salesman problem several branch and bound algorithms for the exact solution of the asymmetric traveling salesman problem atsp based on the assignment problem ap relaxation have been proposed in the literature these algorithms perform very well for some instances e g those with uniformly random integer costs but very poorly for others the aim of this paper is to evaluate the effectiveness of a branch and cut algorithm exploiting atsp specific facet defining cuts to be used to attack hard instances that cannot be solved by the ap based procedures from the literature we present new separation algorithms for some classes of facet defining cuts and a new variable pricing technique for dealing with highly degenerate primal lp problems a branch and cut algorithm based on these new results is designed and evaluated through computational analysis on several classes of both random and real world instances the outcome of the research is that on hard instances the branch and cut algorithm clearly outperforms the best ap based algorithms from the literature

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

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

the circuit polytope facets given an undirected graph g v e and a cost vector c e the weighted girth problem is to find a circuit in g having minimum total cost this problem is in general x1d4a9 x1d4ab hard a promising approach to the solution of hard combinatorial optimization problems is given by the so called cutting plane methods these involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions we consider the weighted girth problem in the case where g is the complete graph kn and study the facial structure of the circuit polytope pcn and some related polyhedra in the appendix we give complete characterizations of pcn for n up to 6 we were motivated to study the weighted girth problem by the fact that this problem and variations of it appear as a subproblem namely the pricing problem in a column generation approach to the vehicle routing problem

worst case comparison of valid inequalities for the tsp we consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst case improvement resulting from their addition to the subtour polyhedron for example we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10 9 the corresponding factor for the class of clique tree inequalities is 8 7 while it is 4 3 for the path configuration inequalities

formulations and valid inequalities for the node capacitated graph partitioning problem we investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition the objective is to minimize the sum of the costs of the edges between the subsets of the partition this problem has a variety of applications for instance in the design of electronic circuits and devices we present alternative integer programming formulations for this problem and discuss the links between these formulations having chosen to work in the space of edges of the multicut we investigate the convex hull of incidence vectors of feasible multicuts in particular several classes of inequalities are introduced and their strength and robustness are analyzed as various problem parameters change

solution of large scale symmetric travelling salesman problems in this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality our implementation is based on a fast lp solver ibm s mpsx and makes effective use of polyhedral results on the symmetric travelling salesman polytope we describe the important ingredients of our code and give an extensive documentation of its computational performance

separation algorithms for classes of stsp inequalities arising from a new stsp relaxation the problem ofseparating a class of inequalities that are valid for the symmetric traveling salesman problem stsp consists of devising an efficient method that given a fractional pointx for a relaxation of the stsp either finds an inequality in the given class of stsp inequalities that is violated byx or determines that there are no such violated inequalities we can define important classes of stsp inequalities by performing naddef s and rinaldi s 1993 node lifting operation on stsp inequalities defined on small graphs we present efficient methods forexactly separating large classes of stsp inequalities that arise from an stsp relaxation related to node lifting in particular we show how to find efficiently the most violated inequality in the class of stsp inequalities having abackbone set with a constant numberk of vertices and thus separate in polynomial time the class of all stsp inequalities whose backbone set has at mostk vertices

an efficient algorithm for the minimum capacity cut problem given a finite undirected graph with nonnegative edge capacities the minimum capacity cut problem consists of partitioning the graph into two nonempty sets such that the sum of the capacities of edges connecting the two parts is minimum among all possible partitionings the standard algorithm to calculate a minimum capacity cut due to gomory and hu 1961 runs in o n4 time and is difficult to implement we present an alternative algorithm with the same worst case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm we report computational results for graphs with up to 2000 nodes

new inequalities for the general routing problem abstract a large new class of valid inequalities is introduced for the general routing problem which properly contains the class of k c constraints these are also valid for the rural postman problem a separation algorithm is given for a subset of these inequalities which runs in polynomial time

efficient separation routines for the symmetric traveling salesman problem ii separating multi handle inequalities this paper is the second in a series of two papers dedicated to the separation problem in the symmetric traveling salesman polytope the first one gave the basic ideas behind the separation procedures and applied them to the separation of comb inequalities we here address the problem of separating inequalities which are all in one way or another a generalization of comb inequalities these are namely clique trees path and ladder inequalities computational results are reported for the solution of instances of the tsplib using the branch and cut framework abacus

new insertion and postoptimization procedures for the traveling salesman problem this paper describes a new insertion procedure and a new postoptimization routine for the traveling salesman problem the combination of the two methods results in an efficient algorithm genius which outperforms known alternative heuristics in terms of solution quality and computing time

a branch and cut algorithm for the equicut problem we describe an algorithm for solving the equicut problem on complete graphs the core of the algorithm is a cutting plane procedure that exploits a subset of the linear inequalities defining the convex hull of the incidence vectors of the edge sets that define an equicut the cuts are generated by several separation procedures that will be described in the paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a branch and cut strategy we also describe the implementation of the algorithm and the interface with the lp solver finally we report on computational results on dense instances with sizes up to 100 nodes

solving multiple knapsack problems by cutting planes in this paper we consider the multiple knapsack problem which is defined as follows given a set n of items with weights f i i in n a set m of knapsacks with capacities f k k in m and a profit function c ik i in n k in m find an assignment of a subset of the set of items to the set of knapsacks that yields minimum cost we consider the multiple knapsack problem from a polyhedral point of view the inequalities that we describe here serve as the theoretical basis for a cutting plane algorithm we present some of the implementation details of this algorithm including a discussion and evaluation of different separation and primal heuristics our algorithm is applied to practical problem instances arising in the design of main frame computers in the layout of electronic circuits and in sugar cane alcohol production

lifted cycle inequalities for the asymmetric traveling salesman problem we investigate the family of facet defining inequalities for the asymmetric traveling salesman ats polytope obtainable by lifting the cycle inequalities we establish several properties of this family that earmark it as arguably the most important among the asymmetric inequalities for the ats polytope known to date in particular we give several results that characterize the pattern of lifting coefficients and thereby facilitate the identification of facet defining inequalities this characterization is used to investigate three new large classes of lifted cycle inequalities that we call shell fork and curtain inequalities these latter inequalities have unbounded chvatal rank furthermore the pattern of lifting coefficients that we identify makes it easy to develop efficient separation routines finally each member of the family is shown to have a counterpart for the symmetric ts sts polytope that is often new and is obtainable by mapping the inequality for the ats polytope into a certain face of the sts polytope and then lifting the resulting inequality into one for the sts polytope itself

a note on the existence and uniqueness of solutions of frequency domain elastic wave problems a priori estimates in h1 abstract in this note we provide existence and uniqueness results for frequency domain elastic wave problems these problems are posed on the complement of a bounded domain r 3 the scatterer the boundary condition at infinity is given by the kupradze sommerfeld radiation condition and involves different sommerfeld conditions on different components of the field our results are obtained by setting up the problem as a variational problem in the sobolev space h 1 on a bounded domain we use a nonlocal boundary condition which is related to the dirichlet to neumann conditions used for acoustic and electromagnetic scattering problems we obtain stability results for the source problem a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences

efficient separation routines for the symmetric traveling salesman problem i general tools and comb separation this is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the traveling salesman problem via branch and cut in this paper we give the basic ideas behind these heuristics and design heuristics for comb separation the second paper will deal with more complex inequalities such as clique tree star or path and ladder inequalities and give computational results

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

a priori optimization of the probabilistic traveling salesman problem the probabilistic traveling salesman problem ptsp is defined on a graph g v e where v is the vertex set and e is the edge set each vertex vi has a probability pi of being present with each edge vi vj is associated a distance or cost cij in a first stage an a priori hamiltonian tour on g is designed the list of present vertices is then revealed in a second stage the a priori tour is followed by skipping the absent vertices the ptsp consists of determining a first stage solution that minimizes the expected cost of the second stage tour the problem is formulated as an integer linear stochastic program and solved by means of a branch and cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function problems involving up to 50 vertices are solved to optimality

separating maximally violated comb inequalities in planar graphs the traveling salesman problem tsp is a benchmark problem in combinatorial optimization it was one of the very first problems used for developing and testing approaches to solving large integer programs including cutting plane algorithms and branch and cut algorithms much of the research in this area has been focused on finding new classes of facets for the tsp polytope and much less attention has been paid to algorithms for separating from these classes of facets in this paper we consider the problem of finding violated comb inequalities if there are no violated subtour constraints in a fractional solution of the tsp a comb inequality may not be violated by more than 1 given a fractional solution in the subtour elimination polytope whose graph is planar we either find a violated comb inequality or determine that there are no comb inequalities violated by 1 our algorithm runs in on mcn time where mcn is the time to compute a cactus representation of all minimum cuts of a weighted planar graph on n vertices

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

relaxed tours and path ejections for the traveling salesman problem abstract we describe an edge based ejection chain method to generate compound neighborhood structures for the traveling salesman problem tsp these neighborhood structures enclose a special substructure which is not necessarily a hamiltonian tour instead the neighborhood components are linked together to compose successive levels of an ejection chain and coordinated by a suitable reference structure to generate compound moves with outstanding proprieties more precisely such a substructure can be viewed as a relaxed tour which allows solution transformations to be obtained without preserving the hamiltonian property at each step furthermore in the ejection chain process the generation of substructures produces a variety of alternating paths for the selection of subsequent ejection moves as well as for the choice of the corresponding trial moves finally we propose two algorithmic variants a preliminary and a full subpath ejection chain method p sec and f sec based on this type of compound neighborhood design both variants are guided by a simple tabu search mechanism computational results on 67 tsplib problems show that both the preliminary and the full methods find optimal and near optimal solutions very quickly frequently outperforming the best heuristic procedures for the tsp in addition the full method generally dominates the best alternative tsp procedures

primal separation for 0 1 polytopes the 0 1 primal separation problem is given an extreme point x of a 0 1 polytope p and some point x find an inequality which is tight at x violated by x and valid for p or assert that no such inequality exists it is known that this separation variant can be reduced to the standard separation problem for p

the black and white traveling salesman problem the black and white traveling salesman problem bwtsp is defined on a graph g whose vertex set is partitioned into black and white vertices the aim is to design a shortest hamiltonian tour on g subject to cardinality and length constraints both the number of white vertices as well as the length of the tour between two consecutive black vertices are bounded above the bwtsp has applications in airline scheduling and in telecommunications this paper proposes an integer linear formulation for the undirected bwtsp as well as several classes of valid inequalities an exact branch and cut algorithm is then developed extensive tests show that it can solve exactly instances involving up to 100 vertices the algorithm can also be applied directly to solve unit demand vehicle routing problems of similar sizes

the sequential knapsack polytope in this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible the corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting

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

separating a superclass of comb inequalities in planar graphs many classes of valid and facet inducing inequalities are known for the family of polytopes associated with thesymmetric travelling salesman problem stsp includingsubtour elimination 2 matching andcomb inequalities for a given class of inequalities anexact separation algorithm is a procedure which given an lp relaxation vectorx finds one or more inequalities in the class which are violated byx or proves that none exist such algorithms are at the core of the highly successfulbranch and cut algorithms for the stsp however whereas polynomial time exact separation algorithms are known for subtour elimination and 2 matching inequalities the complexity of comb separation is unknown apartial answer to the comb problem is provided in this paper we define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx e 0 is planar the separation algorithm runs in o n3 time wheren is the number of vertices in the graph

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

the attractive traveling salesman problem in the attractive traveling salesman problem the vertex set is partitioned into facility vertices and customer vertices a maximum profit tour must be constructed on a subset of the facility vertices profit is computed through an attraction function every visited facility vertex attracts a portion of the profit from the customer vertices based on the distance between the facility and customer vertices and the attractiveness of the facility vertex a gravity model is used for computing the profit attraction the problem is formulated as an integer non linear program a linearization is proposed and strengthened through the introduction of valid inequalities and a branch and cut algorithm is developed a tabu search algorithm is also implemented computational results are reported

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported