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

intrinsic tetrahedron formation of reduced attitude abstract in this paper formation control for reduced attitude is studied in which a regular tetrahedron formation can be achieved and shown to be asymptotically stable under a large family of gain functions in the control moreover by further restriction on the control gain almost global stability of the desired formation is obtained in addition the control proposed is an intrinsic protocol that only uses relative information and does not need to contain any information of the desired formation beforehand the constructed formation pattern is totally attributed to the geometric properties of the space and the designed inter agent connection topology besides a novel coordinates transformation is proposed to represent the relative reduced attitudes in s 2 which is shown to be an efficient approach to reduced attitude formation problems

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

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

new results on the windy postman problem in this paper we study the windy postman problem wpp this is a well known arc routing problem that contains the mixed chinese postman problem mcpp as a special case we extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the windy postman problem over graphs with up to four vertices and ten edges we introduce two new families of facet inducing inequalities and prove that these inequalities along with the already known odd zigzag inequalities are chvatal gomory inequalities of rank at most 2 moreover a branch and cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented finally the performance of a branch and cut algorithm over several sets of large wpp and mcpp instances with up to 3 000 nodes and 9 000 edges and arcs in the mcpp case shows that to our knowledge this is the best algorithm to date for the exact resolution of the wpp and the mcpp

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

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

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

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

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

binary clutter inequalities for integer programs we introduce a new class of valid inequalities for general integer linear programs called binary clutter bc inequalities they include the 0 frac 1 2 cuts of caprara and fischetti as a special case and have some interesting connections to binary matroids binary clutters and gomory corner polyhedra we show that the separation problem for bc cuts is strongly hard in general but polynomially solvable in certain special cases as a by product we also obtain new conditions under which 0 frac 1 2 cuts can be separated in polynomial time these ideas are then illustrated using the traveling salesman problem tsp as an example this leads to an interesting link between the tsp and two apparently unrelated problems the t join and max cut problems

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

cutting planes and the elementary closure in fixed dimension the elementary closurep of a polyhedronp is the intersection ofp with all its gomory chvatal cutting planes p is a rational polyhedron provided thatp is rational the known bounds for the number of inequalities definingp are exponential even in fixed dimension we show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension ifp is a simplicial cone we construct a polytopeq whose integral elements correspond to cutting planes ofp the vertices of the integer hullq iinclude the facets ofp a polynomial upper bound on their number can be obtained by applying a result of cook et al 1992 finally we present a polynomial algorithm in varying dimension which computes cutting planes for a simplicial cone that correspond to vertices ofq i

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

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

tkk algebras and vertex operator representations let be the jordan algebra constructed from a semilattice s of v 1 see allison et al mem amer math soc 603 1997 let be the lie algebra obtained from the jordan algebra by the tits kantor koecher construction the tkk algebra is defined to be the universal central extension of the lie algebra in this paper we present a complete description of the tkk algebra where s is the smallest possible nonlattice semilattice which then allows us to give a representation to this lie algebra by vertex operators

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

an intrinsic approach to formation control of regular polyhedra for reduced attitudes this paper addresses formation control of reduced attitudes in which a continuous control protocol is proposed for achieving and stabilizing all regular polyhedra also known as platonic solids un

structure and representation for a class of infinite dimensional lie algebras abstract let a be a commutative associative algebra over the complex field c and g be the complexification of the real lie algebra so 3 for any fixed elements e 1 e 2 e 3 a we define a lie algebra l e 1 e 2 e 3 g a with lie bracket given by 1 2 when the associative algebra a is the laurent polynomial algebra c t 1 1 t 1 we determine its derivation lie algebra der l and universal central extension l we also give a vertex operator representation for the lie algebra l this new class of lie algebras includes the affine lie algebra a 1 1 and the toroidal lie algebras of type a 1 we note that in general this kind of lie algebras is not z graded

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

intrinsic reduced attitude formation with ring inter agent graph this paper investigates the reduced attitude formation control problem for a group of rigid body agents using feedback based on relative attitude information under both undirected and directed cyc

almost global consensus on the n sphere this paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multiagent systems that evolve in continuous time on the n dimensional unit sphere or n sphere for any connected undirected graph and all n in mathbb n backslash lbrace 1 rbrace each protocol in said class is shown to yield almost global consensus the feedback laws are negative gradients of lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol this convergence result sheds new light on the general problem of consensus on riemannian manifolds the n sphere for n in mathbb n backslash lbrace 1 rbrace differs from the circle and mathsf so 3 where the corresponding protocols fail to generate almost global consensus moreover we derive a novel consensus protocol on mathsf so 3 by combining two almost globally convergent protocols on the n sphere for n in lbrace 1 2 rbrace theoretical and simulation results suggest that the combined protocol yields almost global consensus on mathsf so 3

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

subset row inequalities applied to the vehicle routing problem with time windows this paper presents a branch and cut and price algorithm for the vehicle routing problem with time windows the standard dantzig wolfe decomposition of the arc flow formulation leads to a set partitioning problem as the master problem and an elementary shortest path problem with resource constraints as the pricing problem we introduce the subset row inequalities which are chvatal gomory rank 1 cuts based on a subset of the constraints in the master problem applying a subset row inequality in the master problem increases the complexity of the label setting algorithm used to solve the pricing problem because an additional resource is added for each inequality we propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step like structure of the objective function of the pricing problem computational experiments have been performed on the solomon benchmarks where we were able to close several instances the results show that applying subset row inequalities in the master problem significantly improves the lower bound and in many cases makes it possible to prove optimality in the root node

derivations and central extensions of finitely generated graded lie algebras abstract some general results concerning derivations of finitely generated lie algebras are established these are employed in order to determine the derivations and central extensions of kac moody lie algebras

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

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

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

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

exact algorithm for minimising the number of setups in the one dimensional cutting stock problem the cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste because changing over from one cutting pattern to another involves significant setups an auxiliary problem is to minimise the number of different patterns that are used the pattern minimisation problem is significantly more complex but it is of great practical importance in this paper we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns each of which corresponds to selecting a fixed number of copies of a specific cutting pattern the integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems at each node of the branch and bound tree the linear programming relaxation of our formulation is made tighter by adding super additive inequalities branching rules are presented that yield a balanced tree incumbent solutions are obtained using a rounding heuristic the resulting branch and price and cut procedure is used to produce optimal or approximately optimal solutions for a set of real life problems